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ATORY 
ASTRONOMY 


WILLSON 


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LABORATORY  ASTRONOMY 


BY 


ROBERT  WHEELER  WLLLSON,  PH.D. 

PROFESSOR  OF  ASTRONOMY  IN  HARVARD  UNIVERSITY 


GINN  &  COMPANY 

BOSTON  •  NEW  YORK  •  CHICAGO  •  LONDON 


COPYRIGHT,  1900, 1905 
BY  ROBERT  W.  WILLSON 


ALL  RIGHTS   RESERVED 
66.1 

ASTROHOHT  DEPT. 


gtftenaeum 


GINN   &   COMPANY  •   PRO- 
PRIETORS •  BOSTON  •  U.S.A. 


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PREFACE 


THE  subjects  treated  in  elementary  text-books  of  astronomy  which 
are  most  difficult  and  discouraging  to  the  beginner  are  those  which 
deal  with  the  diurnal  motion  of  the  heavens  and  the  apparent 
motions  of  the  sun,  moon,  and  planets  among  the  stars.  A  clear 
conception  of  these  fundamental  facts  is,  however,  necessary  to 
a  proper  understanding  of  many  of  the  striking  phenomena  to 
which  the  study  of  astronomy  owes  its  hold  upon  the  intellect  and 
the  imagination. 

No  adequate  notion  of  those  subjects  which  involve  the  ideas  of 
force  and  mass  can  be  given  to  the  average  student  who  has  not 
mastered  the  elements  of  mechanics ;  but  to  explain  the  motions 
of  the  heavenly  bodies,  the  knowledge  of  a  few  principles  of  solid 
geometry  and  of  the  properties  of  the  ellipse  will  suffice,  —  no 
more,  indeed,  than  may  "be  easily  explained-  in  the  pages  of  the 
text-book  itself. 

Most  of  the  difficulties  which  arise  at  the  outset  of  the  study  may 
be  satisfactorily  met  by  methods  which  require  the  student  to  make 
and  discuss  simple  observations  and  to  solve  simple  problems.  This 
necessity  is  recognized  in  many  recent  text-book's  which  introduce 
such  methods  to  a  greater  or  less  extent,  —  in  all  cases  to  great 
advantage  and  in  some  with  marked  success.  I  have  gathered  in 
this  book  some  of  those  which  I  have  found  practicable,  intending 
that  they  should  explain  in  natural  sequence  those  phenomena  which 
depend  on  the  diurnal  motion,  the  moon's  motion  in  her  orbit  and 
the  change  in  position  of  that  orbit,  the  motion  of  the  sun  in  the 
ecliptic,  and  the  geocentric  motions  of  the  planets. 

The  methods  chosen  may  be  carried  out  with  fair-sized  classes 
and  do  not  require  a  place  of  observation  favored  with  an  extensive 
view  of  the  heavens.  The  gnomon-pin,  the  hemisphere,  the  cross- 
staff,  a  simple  apparatus  for  measuring  altitude  and  azimuth  which 

iii 


iv  PREFACE 

may  be  converted  into  an  equatorial  by  inclining  it  at  the  proper 
angle,  together  with  a  few  maps  and  diagrams,  form  an  outfit  so 
inexpensive  that  it  may  be  supplied  to  each  pupil,  and  much  work 
may  be  done  at  home.  It  is  obvious  that  the  possibility  thus  offered 
of  utilizing  favorable  opportunities  for  observation  is  especially 
valuable  in  a  study  which  is  so  much  dependent  on  the  weather. 
All  members  of  the  class,  too,  will  be  doing  the  same  or  similar 
work  at  the  same  time,  —  a  principle  of  cardinal  importance  in 
elementary  laboratory  work  with  large  classes. 

The  meridian  work  of  Chapter  VI  is  added  for  the  sake  of  logical 
completeness,  to  explain  the  determination  of  the  zero  of  right 
ascensions,  —  a  subject  which  is  usually  neglected  in  the  text-books 
and  would  not  be  included  in  an  ordinary  course. 

Nothing  has  been  directly  planned  for  teaching  the  names  of  the 
constellations  and  the  use  of  star  maps.  The  work  of  Chapters  II, 
III,  and  IV,  covering  a  period  of  some  months,  results  in  a  very 
good  acquaintance  with  the  principal  stars  and  asterisms.  It  may 
be  assumed,  too,  that  the  teacher  is  familiar  with  the  heavens  and 
will  gather  the  class  as  early  as  possible  to  introduce  them  at  least 
to  the  polar  constellations. 

The  book  is  intended  primarily  for  teachers,  but  much  of  it  is 
suitable  for  use  as  a  text-book,  in  spite  of  its  rather  condensed 
form.  It  is  meant  to  be  used  in  connection  with  one  of  the  many 
admirable  text-books  on  descriptive  astronomy  adapted  to  high- 
school  pupils. 

The  first  six  chapters  were  printed  in  1900,  and  various  changes 
and  additions  might  now  be  made,  notably  an  improvement  in  the 
protractor  for  laying  off  altitudes  on  the  hemisphere,  which  is  now 
so  constructed  that  it  may  be  used  as  a  ruler  for  the  accurate  draw- 
ing of  great  circles.  This  permits  a  much  simpler  determination  of 
the  pole  of  a  small  circle  than  that  described  in  the  first  chapter. 

ROBERT   W.   WILLSON 
HARVARD  UNIVERSITY 
STUDENTS'  ASTRONOMICAL  LABORATORY 
December,  1905 


TABLE   OF   CONTENTS 

CHAPTER  I 

THE   SUN'S   DIURNAL   MOTION 

PAGE 

Path  of  the  Shadow  of  a  Pin-head  cast  by  the  Sun  upon  a  Horizontal  Plane  1 

Altitude  and  Bearing        ..........  4 

Representation  of  the  Celestial  Sphere  upon  a  Spherical  Surface.        .        .  6 
The  Sun's  Diurnal  Path  upon  the  Hemisphere  is  a  Circle  —  a  Small  Circle 

except  about  March  20  and  September  21 8 

Determination  of  the  Pole  of  the  Circle    .......  9 

Bearing  of  the  Points  of  Sunrise  and  Sunset 11 

The  Meridian  —  the  Cardinal  Points 11 

Magnetic  Decimation 12 

Azimuth  ..............  12 

The  Equinoctial   ............  14 

Position  of  the  Pole  as  seen  from  Different  Places  of  Observation    .         .  15 

Latitude  equals  Elevation  of  Pole 16 

Hour-angle  of  the  Sun      ..........  17 

Uniform  Increase  of  the  Sun's  Hour-angle  —  Apparent  Solar  Time     .         .  18 
Declination  of  the  Sun  —  its  Daily  Change       .        .        .         .        .        .20 

CHAPTER   II 

THE   MOON'S   PATH   AMONG   THE    STARS 

Position  of  the  Moon  by  its  Configuration  with  Neighboring  Stars       .         .21 

Plotting  the  Position  of  the  Moon  upon  a  Star  Map          ....  24 

Position  of  the  Moon  by  Measures  of  Distance  from  Neighboring  Stars        .  25 

The  Cross-staff 25 

Length  of  the  Month ' 29 

Node  of  the  Moon's  Orbit 30 

Errors  of  the  Cross-staff                                                                                       .  31 


VI  TABLE    OF    CONTENTS 

CHAPTER    III 

THE   DIURNAL   MOTION    OF   THE    STARS 

PAGE 

Instrument  for  measuring  Altitude  and  Azimuth 34 

Adjustment  of  the  Altazimuth .35 

Determination  of  Meridian  by  Observations  of  the  Sun  ....  37 
Determination  of  Apparent  Noon  by  Equal  Altitudes  of  the  Sun  .  .  39 

Meridian  Mark 40 

Selection  of  Stars  —  Magnitudes  .         .         .         .         .         .         .         .         .41 

Plotting  Diurnal  Paths  of  Stars  on  the  Hemisphere          ....         42 

Paths  of  Stars  compared  with  that  of  the  Sun .42 

Drawing  of  Hemisphere  with  its  Circles  .......         42 

Rotation  of  the  Sphere  as  a  Whole 43 

Declinations  of  Stars  do  not  change  like  that  of  the  Sun          ...         43 
Equable  Description  of  Hour-angle  by  Stars         ......     43 

Hour-angle  and  Declination  fix  the  Position  of  a  Heavenly  Body  as  well 
as  Altitude  and  Azimuth  —  Comparison   of  the  Two   Systems  of 
Coordinates    ...........         44 

Equatorial  Instrument  for  measuring  Hour-angle  and  Declination  .  .  45 
Universal  Equatorial  —  Advantages  of  the  Equatorial  Mounting  .  .  45 

CHAPTER   IV 

THE  COMPLETE  SPHERE  OF  THE  HEAVENS 

Rotation  of  the  Heavens  about  an  Axis  passing  through  the  Pole  explains 

Diurnal  Motions  of  Sun,  Moon,  and  Stars          .         .         .         .         .47 
Relative  Position  of  Two  Stars  determined  by  their  Declinations  and  the 

Difference  of  their  Hour-angles 48 

Use  of  Equatorial  to  determine  Positions  of  Stars  .  .  .  .  .49 
Use  of  a  Timepiece  to  improve  the  Foregoing  Method  ....  50 
Map  of  Stars  by  Comparison  with  a  Fundamental  Star  .  .  .  .53 
Extension  of  Use  of  Timepiece  to  reduce  Labor  of  Observation  .  .  54 
The  Vernal  Equinox  to  replace  the  Fundamental  Star  —  Right  Ascension  56 

Sidereal  Time  —  Sidereal  Clock 57 

Right  Ascension  of  a  Star  is  the  Sidereal  Time  of  its  Passage  across  the 

Meridian 58 

Right  Ascension  of  any  Body  plus  its  Hour-angle  at  any  Instant  is  Side- 
real Time  at  that  Instant        ........         58 

Finding  Stars  by  the  Use  of  a  Sidereal  Clock  and  the  Circles  of  the  Equa- 
torial Instrument        ..........     59 

The  Clock  Correction 60 

List  of  Stars  for  determining  Clock  Error 61 


TABLE  OF  CONTENTS  vii 

CHAPTER   V 

MOTION  OF  THE  MOON  AND   SUN  AMONG  THE  STARS 

PAGE 

Plotting  Stars  upon  a  Globe  in  their  Proper  Relative  Positions         .         .  63 
Plotting  Positions  of  the  Moon  upon  Map  and  Globe  by  Observations  of 
Declination,  and  Difference  of  Right  Ascension,  from  Neighboring 

Stars '       .         .         .         .         .64 

Variable  Rate  of  Motion  of  the  Moon       .         .         .         . '       .         .         .  65 

Variable  Semi-diameter  of  the  Moon    ........  65 

Position  of  Greatest  Semi-diameter  and  of  Greatest  Angular  Motion        .  (  65 

Plotting  Moon's  Path  on  an  Ecliptic  Map 65 

Observations  of  Sun's  Place  in  Reference  to  a  Fundamental  Star  by  Equa- 
torial and  Sidereal  Clock        ........  66 

Sun's  Place  referred  to  Stars  by  Comparison  with  the  Moon  or  Venus         .  68 

Plotting  the  Sun's  Path  upon  the  Globe  —  the  Ecliptic     ....  70 

CHAPTER  VI 

MERIDIAN  OBSERVATIONS 

Use  of  the  Altazimuth  or  Equatorial  in  the  Meridian 72 

The  Meridian  Circle 73 

Adjustments  of  the  Meridian  Circle     ........  74 

Level        .............  74 

Collimation  .............  78 

Azimuth .         ...  78 

Determination  of  Declinations      .         . 80 

Determination  of  the  Polar  Point      .         .         .         .         .         .         .         .  81 

Absolute  Determination  of  Declination         .......  81 

Determination  of  the  Equinox 83 

Absolute  Right  Ascensions  .         ...         .         .         .         .         .         .         .84 

Autumnal  Equinox  of  1899 85 

Autumnal  Equinox  of  1900 *  .         .         .87 

Length  of  the  Year  .                           88 

CHAPTER  VII 

THE  NAUTICAL  ALMANAC 

Mean  Time 91 

The  Equation  of  Time 92 

Standard  Time    .  93 


viii  TABLE   OF  CONTENTS 

PAGE 

The  Calendar  Pages 94 

Examination  of  the  Several  Columns          .......  99 

Data  for  the  Planets  and  Stars        ........  102 

Comparison  of  Observations  with  the  Ephemeris 102 

Observations  of  the  Moon  with  the  Cross-staff ;  Length  of  the  Month    .  103 

Observation  at  Apparent  Noon 104 

Observations  of  the  Planets.    Observations  of  the  Moon  with  Equatorial  105 

Observations  of  the  Sun's  Place          ........  106 

Determination  of  the  Equinox         ........  107 


CHAPTER  VIII 
THE  CELESTIAL,  GLOBE 

Description  of  the  Globe     .         »        .        .         ,      "  .         .         .         .         .  .108 

Rectifying  the  Globe  for  a  Given  Place  and  Time  .        .         .         .         .  Ill 

The  Sun's  Place  on  the  Globe     .         .         .       :,.      .       ...         .         .112 

The  Altitude  Arc 113 

Problems  which  do  not  require  Rectification  of  the  Globe           .         .         .  114 

Problems  which  require  Rectification  of  the  Globe  for  a  Given  Time      .  117 
Finding  an  Hour-angle  by  the  Globe           .         .         .         .         .         .         .119 

Reduction  to  the  Equator       .         .         .         .         .         .         .         .  121 

CHAPTER  IX 

EXAMPLES  OF  THE  USE  OF  THE  GLOBE 

Problems  which  require  Rectification  of  the  Globe  for  a  Given  Place     .  122 

Rising  and  Setting  of  Stars 122 

Sunrise    .        . .         .  124 

Altitude  and  Azimuth  ;  Hour-angle    ........  125 

Finding  the  Time  from  the  Sun's  Altitude 126 

Identifying  a  Heavenly  Body  by  its  Altitude  and  Azimuth  at  a  Given  Time  129 

Aspect  of  the  Planets  at  a  Given  Time    .......  130 

Rising  and  Setting  of  the  Moon 131 

Twilight 133 

Orientation  of  Building  by  Sun  Observation 134 

Latitudes  in  which  Southern  Cross  is  Visible          .         .         .         ...  135 

The  Midnight  Sun  ;  the  Harvest  Moon 136 

Change  of  Azimuth  at  Rising  and  Setting 137 

Graduating  a  Horizontal  Sundial .         .  137 

Graduating  a  Vertical  Sundial 138 

Determining  Path  of  Shadow  by  Globe 139 

The  Hour-index  141 


TABLE  OF  CONTENTS  IX 

CHAPTER  X 

THE  MOTION  OF  THE  PLANETS 

PAGE 

Elliptic  Orbits  a  Result  of  the  Law  of  Gravitation      .....     143 

Properties  of  the  Ellipse 144 

To  draw  an  Ellipse  from  Given  Data          .         .         . '  '     '.        .        .         .145 
Mean  and  True  Place  of  a  Planet ;  Equable  Description  of  Areas          .          146 

The  Equation  of  Center .         .         .148 

Measurement  of  Angles  in  Radians         . 149 

The  Diagram  of  Curtate  Orbits .151 

To  find  the  Elements  of  an  Orbit  from  the  Diagram       .         .         .         .         154 

Place  of  the  Planet  in  its  Orbit -     .         .156 

To  find  the  True  Heliocentric  Longitude  of  a  Planet      .         .         .         .         157 

To  find  the  Heliocentric  Latitude        ........     161 

Geocentric  Longitude  of  a  Planet   .         .         .         .         ...         .         .         161 

The  Sun's  Longitude  and  the  Equation  of  Time 162 

Geocentric  Latitude        ..........         163 

Perturbations;  Precession 166 

The  Julian  Day 167 

Right  Ascensions  and  Declinations  of  the  Planets 167 

Configurations  of  the  Planets 168 

The  Path  of  Mars  among  the  Stars  in  1907 169 


LABORATORY  ASTRONOMY 


PART  I 

CHAPTER   I 
THE  DIURNAL  MOTION  OF  THE  SUN 

THE  most  obvious  and  important  astronomical  phenomenon  that 
men  observe  is  the  succession  o.f  day  and  night,  and  the  motion  of 
the  sun  which  causes  this  succession  is  naturally  the  first  object  of 
astronomical  study.  Every  one  knows  that  the  sun  rises  in  the  east 
and  sets  in  the  west,  but  very  many  educated  people  know  little 
more  of  the  course  of  the  sun  than  this.  The  first  task  of  the 
beginner  in  astronomy  should  be  to  observe,  as  carefully  as  possible, 
the  motion  of  the  sun  for  a  day.  What  is  to  be  observed  then? 
A  little  thought  shows  that  it  can  only  be  the  direction  in  which 
we  have  to  look  to  see  it  at  different  times ;  that  is,  toward  what 
point  of  the  compass  —  how  far  above  the  ground.  All  astronom-  / 
ical  observation,  indeed,  comes  down  ultimately  to  this  —  the  direc-  / 
tion  in  which  we  see  things.  The  strong  light  of  the  sun  enables 
us  to  make  use  of  a  very  simple  method  depending  on  the  principl 
that  the  shadow  of  a  body  lies  in  the  same  straight  line  with 
body  and  the  source  of  light. 

Path  of  the  Shadow  of  a  Pin-head.  —  If  we  place  a  pin  upright  on 
a  horizontal  plane  in  the  sunlight  and  mark  the  position  of  the 
shadow  of  its  head  at  any  time,  we  thus  fix  the  position  of  the 
sun  at  that  time,  since  it  is  in  the  prolongation  of  the  line  drawn 
from  the  shadow  to  the  pin-head.  In  order  to  carry  out  systematic 

1 


enables  *^ 

?inciple  /  0#« 
ith  the/  ^ 


2  LABORATORY   ASTRONOMY 

observations  by  this  method  in  such  a  form  that  the  results  may  be 
easily  discussed,  it  will  be  convenient  to  have  the  following  appa- 
ratus :  (1)  A  firm  table  in  such  a  position  as  to  receive  sunlight  for 
as  long  a  period  as  possible.  It  is  better  that  it  should  be  in  the 
open  air,  in  which  case  it  may  be  made  by  driving  small  posts  into 


FIG.  i 

the  ground  and  securely  fastening  a  stout  plank  about  18  inches 
square  as  a  top.  (2)  A  board,  18  inches  long  and  8  inches  broad, 
furnished  with  leveling  screws  and  smoothly  covered  with  white 
paper  fastened  down  by  (3)  thumb  tacks.  (4)  A  level  for  leveling 
the  board.  (5)  A  compass.  (6)  A  glass  plate,  6  inches  long  and 
2  inches  broad,  along  the  median  line  of  which  a  straight  black 
line  is  drawn.  (7)  A  pin,  5  cm.  long,  with  a  spherical  head  and 
an  accurately  turned  base  for  setting  it  vertical.  (8)  A  timepiece. 
Draw  a  straight  pencil  line  across  the  center  of  the  paper  as 


FIG.  2 

nearly  as  possible  perpendicular  to  the  length  of  the  board.  Place 
the  board  upon  the  table  and  level  approximately.  Put  the  com- 
pass on  the  middle  of  the  pencil  line  and  put  the  glass  plate  on  the 
compass  with  its  central  line  over  the  center  of  the  needle ;  turn 
the  plate  till  its  median  line  is  parallel  to  the  pencil  line  (Fig.  2), 


THE    DIURNAL    MOTION    OF    THE    SUN  3 

and  swing  the  whole  board  horizontally,  till  the  needle  is  parallel  to 
the  two  lines,  which  are  then  said  to  be  in  the  magnetic  meridian. 
Press  the  leveling  screws  firmly  into  the  table,  and  thus  make  dents 
by  which  the  board  may  at  any  future  time  be  placed  in  the  same 
position  without  the  renewed  use  of  the  compass.  Level  the  board 


carefully,  placing  the  level  first  east  and  west,  then  north  and  south. 
Place  the  pin  in  the  pencil  line, — in  the  center  if  the  observation 
is  made  between  March  20  and  September  20,  but  near  the  south- 
ern edge  of  the  board  at  any  other  time  of  the  year,  —  pressing  it 
firmly  down  till  the  base  is  close  to  the  paper,  so  that  the  pin  is 
perpendicular  to  the  paper.  Mark  with  a  hard  pencil  the  estimated 
center  of  the  shadow  of  the  pin-head,  A  (Fig.  3),  noting  the  time  by 
the  watch  to  the  nearest  minute,  affix  a  number  or  letter,  and  affix 
the  same  number  to  the  recorded  time  of  the  observation  in  the 
note-book.  It  is  a  good  plan  to  use  pencil  for  notes  made  while 


FIG.  4 


observing,  and  ink  for  computations  or  notes  added  afterward  in 
discussing  them.  Kepeat  at  hourly,  or  better  half -hourly,  intervals, 
thus  fixing  a  set  of  points  (Fig.  4),  through  which  a  continuous 
curve  may  be  drawn  showing  the  path  of  the  shadow  for  several 
hours.  The  same  observation  should  be  repeated  two  weeks  later. 


LABORATORY    ASTRONOMY 


ALTITUDE    AND   BEARING 

By  the  foregoing  process  we  obtain  a  diagram  on  which  is  shown 
the  position  of  the  pin  point,  a  magnetic  meridian  line  through  this 
point,  and  a  series  of  numbered  points  showing  the  position  of  the 
shadow  of  the  pin-head  at  different  times ;  the  height  of  the  pin  is 
known  and  also  the  fact  that  its  head  was  in  the  same  vertical  line 
with  its  point. 

In  the  discussion  of  these  results,  it  will  be  convenient  to  proceed 
as  follows  : 

Eemove  the  pin  and  draw  with  a  hard  pencil  a  fine  line,  AB 
(Fig.  5),  through  the  pinhole  and  the  point  marked  at  the  first  obser- 
vation. This  line  is  called  a  line  of  bearing,  and  the  angle  which 


FIG.  5 


it  makes  with  the  magnetic  meridian  is  called  the  magnetic  bearing 
of  the  line.  This  angle,  which  may  be  directly  measured  on  the 
diagram  by  a  protractor,  fixes  the  position  of  the  vertical  plane  which 
contains  the  observed  point  and  passes  also  through  the  center  of 
the  pin-head  and  the  sun.  If  this  point  bears  KW.  from  the  pin, 
the  sun  evidently  bears  S.E. 

Imagine  a  line,  AC  (Fig.  3),  connecting  the  observed  point  with  the 
sun's  center  and  passing  also  through  the  center  of  the  pin-head. 
The  position  of  the  sun  in  the  vertical  plane  is  evidently  fixed  by 
this  line.  The  angle  between  the  line  of  bearing  and  this  line,  BA  C, 
is  called  the  altitude  of  the  sun ;  it  measures,  by  the  ordinary  con- 
vention of  solid  geometry,  the  angle  between  the  sun's  direction 
and  the  plane  of  the  horizon. 


THE  DIURNAL  MOTION  OF  THE  SUN          o 

To  determine  this  angle,  lay  off  the  line  B'C'  (Fig.  6),  equal  in 
length  to  the  pin,  5  cm.,  draw  a  perpendicular  through  E\  and  by 
means  of  a  pair  of  compasses  or  scale  laid 
between  the  two  points  A  and  B  (Fig.  5), 
lay  off  the  line  A'B'  on  the  perpendicular, 
draw  A'C'f  and  measure  the  angle  B'A'C' 
by  a  protractor.  We  now  have  the  bearing 
and  altitude  of  the  sun  at  the  time  of  the 
first  observation,  the  bearing  of  the  sun 
from  the  pin  being  opposite  to  that  of 
the  point  from  the  pin.  In  like  manner 
the  altitude  and  bearing  are  determined  for 
each  observed  point  upon  the  path  of  the 
shadow,  and  noted  against  the  correspond- 
ing time,  in  the  note-book  (to  avoid  con- 
fusion, it  is  convenient  to  make  a  separate 
figure  for  the  morning  and  afternoon 
observations,  as  shown  in  Fig.  6).  We 
have  thus  obtained  a  series  of  values 
which  will  enable  us  to  study  more  easily 
the  path  of  the  sun  upon  the  concave  of 
the  sky. 

Plotting  the  Sun's  Path  on  a  Spherical  Surface.  —  Probably   the 
most   evident  method  of  accomplishing  this  object  would   be  to 


construct  a  small  concave  portion  of  a  sphere,  as  in  the  accom- 
panying figure,  which  suggests  how  the  position  of  the  sun  might 
be  referred  to  the  inside  of  a  glass  shell. 


6  LABORATORY    ASTRONOMY 

But  the  hollow  surface  offers  difficulty  in  construction  and 
manipulation,  and  it  requires  but  little  stretch  of  the  imagination 
to  pass  to  the  convex  surface  as  follows.  The  glass  shell,  as 
seen  from  the  other  side,  would  appear  thus : 


FIG. 


and  we  can  more  readily  get  at  it  to  measure  it,  and  moreover  can 
more  easily  recognize  the  properties  of  the  lines  which  we  shall 
come  to  draw  upon  it,  since  we  are  used  to  looking  upon  spheres 
from  the  outside  rather  than  from  the  inside,  except  in  the  case  of 
the  celestial  sphere. 

On  both  Figs.  7  and  8  is  shown  a  group  of  dots  which  have 
nearly  the  configuration  of  a  group  of  stars  conspicuous  in  the 
southern  heavens  in  midsummer  and  called  the  constellation  of 
Scorpio.  It  is  evident  that  the  constellation  has  the  same  shape  in 
both  cases,  except  that  in  Fig.  8  it  is  turned  right  for  left  or  semi- 
inverted,  as  is  the  image  of  an  object  seen  in  a  mirror.  This  prop- 
erty obviously  belongs  to  all  figures  drawn  on  the  concave  surface 
as  seen  from  the  center,  when  they  are  looked  at  from  the  outside 
directly  toward  the  center. 

So  also  the  diurnal  motion  of  the  sun,  which  as  we  see  it 
from  the  center  is  from  left  to  right,  would  be  from  right  to 
left  as  viewed  from  the  outside  of  such  a  surface.  This  latter 
is  so  slight  an  inconvenience  that  it  is  customary  to  represent 
the  motions  of  the  heavenly  bodies  in  the  sky  upon  an  opaque 
globe,  and  to  determine  the  angles  which  these  bodies  describe 
about  the  center,  by  measuring  the  corresponding  arcs  upon  the 
convex  surface. 


THE  DIURNAL  MOTION  OF  THE  SUN       7 

Plotting  on  a  Hemisphere.  —  The  apparatus  required  for  plotting 
the  sun's  path  consists  of :  a  hemisphere,,  a,  4{-  inches  in  diameter ; 
a  circular  protractor,  b,  a  quadrantal  protractor,  c,  of  2^-  inches 


FIG.  9 

radius,  and  a  pair  of  compasses,  d,  whose  legs  may  be  bent  and  one 
of  which  carries  a  hard  pencil  point. 

Determine  by  trial  with  the  compasses  the  center  of  the  base  of 
the  hemisphere,  and  mark  two  diameters  by  drawing  straight  lines 
upon  the  base  at  right  angles  through  the  center.  Prolong  these  by 
marks  about  -J  inch  in  length  upon  the  convex  surface.  Place  the 


Fio.  10 


hemisphere  exactly  central  upon  the  circular  protractor,  by  bring- 
ing the  marked  ends  of  one  of  the  diameters  upon  those  divisions 
of  the  protractor  which  are  numbered  0°  and  180°,  and  the  other  on 
the  divisions  numbered  90°  and  270°.  Determine  and  mark  the 


8 


LABORATORY    ASTRONOMY 


highest  point  of  the  hemisphere  by  placing  the  quadrant  with  its 
base  upon  the  circular  protractor,  and  its  arc  closely  against  the 
sphere,  and  marking  the  end  of  the  scale  (Fig.  10).  Eepeat  this 
with  the  arc  in  four  positions,  90°  apart  on  the  base.  The  points 
thus  determined  should  coincide ;  if  they  do  not,  estimate  and  mark 
the  center  of  the  four  points  thus  obtained.  This  point  represents 
the  highest  point  of  the  dome  of  the  heavens  —  the  point  directly 
overhead,  called  the  zenith,  and  the  zero  and  180°  points  on  the  base 
protractor  may  be  taken  as  representing  the  south  and  north  points 
respectively  of  the  magnetic  meridian. 

The  Sun's  Path  a  Circle.  —  To  plot  the  altitude  and  bearing  of  the 
first  observation,  place  the  foot  of  the  quadrant  or  altitude  arc 
close  against  the  sphere,  the  foot  of  its  graduated  face  on  the 
degree  of  the  protractor  which  corresponds  to  the  bearing.  Mark 
a  fine  point  on  the  sphere  at  that  degree  of  the  altitude  arc  corre- 
sponding to  the  altitude  at  the  first  observation.  This  point  fixes 
the  direction  in  which  the  sun  would  have  been  seen  from  the  center 
of  the  hemisphere  at  the  time  of  observation  if  the  zero  line  had 
been  truly  in  the  magnetic  meridian.  Proceed  in  the  same  manner 
with  the  other  observations  of  bearing  and  altitude,  and  thus  obtain 


FIG.  11 


a  series  of  points  (Fig.  11),  through  which  may  be  drawn  a  con- 
tinuous line  representing  the  sun's  path  upon  that  day. 

It  will  appear  at  once  that  the  arcs  between  the  successive  points 
are  of  nearly  equal  length  if  the  times  of  observation  were  equi- 
distant, and  otherwise  are  proportional  to  the  intervals  of  time 


THE  DIURNAL  MOTION  OF  THE  SUN 

between  the  corresponding  observations  —  a  property  which  does 
not  at  all  belong  to  the  shadow  curve  from  which  the  points  are 
derived.  We  thus  have  a  noteworthy  simplification  in  referring 
our  observations  to  the  sphere.  It  will  also  appear  that  a  sheet  of 


FIG.  12 

stiff  paper  or  cardboard  may  be  held  edgewise  between  the  hemi- 
sphere and  the  eye,  so  as  to  cover  all  the  points ;  that  is,  they  all 
lie  in  the  same  plane.  This  fact  shows  that  the  sun's  path  is  a 
circle  on  the  sphere.  It  is  shown  by  the  principles  of  solid  geometry 
that  all  sections  of  the  sphere  by  a  plane  are  circles.  If  the  plane 
of  the  circle  passes  through  the  center,  it  is  the  largest  possible,  its 
radius  being  equal  to  that  of  the  sphere ;  it  is  then  called  a  great 
circle.  Near  the  20th  of  March  and  22d  of  September  it  will  be 
found  that  the  path  of  the  shadow  is  nearly  a  straight  line  on  the 
diagram,  and  that  the  path  of  the  sun  is  nearly  a  great  circle ;  that 
is,  the  plane  of  this  circle  passes  nearly  through  the  center  of  the 
sphere.  In  general,  the  shadow  path  is  a  curve,  with  its  concave 
side  toward  the  pin  in  summer  and  its  convex  side  toward  it  in 
winter,  while  the  path  on  the  sphere  is  a  small  circle,  that  is,  its 
plane  does  not  pass  through  the  center  of  the  sphere. 

Determining  the  Pole  of  the  Circle.  —  It  is  proved  by  solid  geometry 
that  all  points  of  any  circle  on  the  sphere  are  equidistant  from  two 


10 


LABORATORY   ASTRONOMY 


points  on  the  sphere,  called  the  poles  of  the  circle.     It  is  important 
to  determine  the  pole  of  the  sun's  diurnal  path. 

Estimate  as  closely  as  possible  the  position  on  the  sphere  of  a 
point  which  is  at  the  same  distance  from  all  the  observed  points  of 
the  sun's  path  and  open  the  compasses  to  nearly  this  distance.  For 
a  closer  approximation  to  the  position  of  the  pole,  place  the  steel 
point  of  the  compasses  at  the  point  on  the  hemisphere  correspond- 
ing to  the  first  observation,  a,  and  with  the  other  (pencil)  point  draw 
a  short  arc,  m  (Fig.  12),  near  the  estimated  pole.  Draw  the  arc  n 

from  the  point  of  the 
last  observation,  <?,  and 
join  these  two  arcs  by 
a  third  drawn  from  an 
observed  point,  b,  as 
near  as  possible  to  the 
middle  of  the  path; 
the  pole  of  the  sun's 
diurnal  circle  will  lie 
nearly  on  the  great 
circle  drawn  from  b  to 
the  middle  point  o  of 
the  arc  last  drawn. 
Place  the  steel  point 
at  o,  and  the  pencil 
point  at  b,  and  try  the 
distance  of  the  pencil 
FIG.  is  point  from  the  sun's 

path  at  either  ex- 
tremity. If  the  pencil  point  lies  above  (or  below)  the  path  at  both 
extremities,  the  compasses  must  be  opened  (or  closed)  slightly  and 
the  assumed  pole  shifted  directly  away  from  (or  toward)  the  middle 
of  the  path. 

The  proper  opening  of  the  compasses  is  thus  quickly  determined 
as  well  as  a  close  approximation  to  the  position  of  the  pole.  Place 
the  steel  point  at  this  new  position,  p,  the  pencil  point  at  b,  and 
again  test  the  extreme  points.  If  the  west  end  of  the  path  is  below 
the  pencil  point  (Fig.  13),  the  latter  should  be  brought  directly  down 


THE    DIURNAL    MOTION    OF    THE    SUN  11 

to  the  path  by  shifting  the  steel  point  on  the  sphere  in  the  plane 
of  the  compass  legs,  that  is,  along  the  great  circle  from  p  to  s. 

From  the  point  thus  found  a  circle  can  be  described  with  the 
compasses  so  as  to  pass  approximately  through  all  the  observed 
points ;  that  is,  this  point  is  the  pole  of  the  sun's  path,  and  when 
it  is  fixed  as  exactly  as  possible  a  circle  is  to  be  drawn  from  horizon 
to  horizon  which  will  represent  the  sun's  path  from  the  point  of 
sunrise  to  that  of  sunset,  and  passing  very  nearly  through  all  the 
observed  points.  The  bearing  of  the  points  of  sunrise  and  sunset 
may  then  be  read  off  on  the  horizontal  circle. 

THE   MERIDIAN 

The  pole  as  thus  determined  marks  a  very  interesting  and 
important  point  in  the  heavens.  We  will  draw  a  great  circle 
through  the  zenith  and  the  pole.  To  do  this,  place  the  altitude  arc 
against  the  sphere,  as  if  to  measure  the  altitude  of  the  pole ;  and 


FIG.  14 

using  it  as  a  guide,  draw  the  northern  quadrant  of  the  vertical 
circle  through  the  zenith  and  the  pole.  Note  the  bearing  of  this 
vertical  circle.  Place  the  altitude  arc  at  the  opposite  bearing,  and 
draw  another  or  southern  quadrant  of  the  same  great  circle  till  it 
meets  the  south  horizon.  This  great  circle  (Fig.  14)  is  called  the 
meridian  of  the  place  of  observation,  and  its  plane  is  called  the 
plane  of  the  meridian  of  the  place  of  observation,  —  sometimes 
the  true  meridian,  to  distinguish  it  from  the  magnetic  meridian. 


12  LABORATORY    ASTRONOMY 

The  line  in  which  it  cuts  the  base  of  the  hemisphere  represents  the 
meridian  line  or  true  meridian  line,  just  as  the  line  first  drawn  repre- 
sents the  line  of  the  magnetic  meridian.  If  the  observations  are  made 
in  the  United  States,  near  a  line  drawn  from  Detroit  to  Savannah, 
it  will  be  found  that  the  true  meridian  coincides  very  nearly  with 
the  magnetic  meridian.  East  of  the  line  joining  these  cities,  the 
north  end  of  the  magnet  points  to  the  west  of  the  true  meridian  by 
the  amounts  given  in  the  following  table : 

21°  at  the  extreme  N.E.  boundary  of  Maine. 
15    at  Portland. 
10    at  Albany  and  New  Haven. 
5    at  Washington  and  Buffalo. 

While  on  the  west  the  declination,  as  it  is  called,  is  to  the  east  of  the 
true  meridian. 

5°  at  St.  Louis  and  New  Orleans. 
10    at  Omaha  and  El  Paso. 
15    at  Deadwood  and  Los  Angeles. 
20    at  Helena,  Montana,  and  C.  Blanco. 
23    at  the  extreme  N.W.  boundary  of  the  United  States. 

By  drawing  these  lines  on  the  map,  as  in  Fig.  15,  it  is  easy  to 
estimate  the  declinations  at  intermediate  points  within  one  or  two 
degrees,  —  at  the  present  time  west  declinations  in  the  United  States 
are  increasing  and  east  declinations  decreasing  by  about  1°  in  fifteen 
years. 

A  great  circle  perpendicular  to  the  meridian  may  be  drawn  by 
placing  the  altitude  protractor  at  readings  90°  and  270°  from  the 
meridian  reading  and  drawing  arcs  to  the  zenith  in  each  case. 
This  circle  is  the  prime  vertical,  and  intersects  the  horizon  in  the 
east  and  west  points ;  thus  all  the  cardinal  points  are  fixed  by  the 
meridian  determined  from  our  plotting  of  the  sun's  path. 

Azimuth.  —  Place  the  hemisphere  upon  the  circular  protractor  in 
such  a  position  that  the  line  of  the  true  meridian  on  the  hemisphere 
coincides  with  the  zero  line  of  the  protractor. 

Place  the  altitude  arc  so  as  to  measure  the  altitude  at  any  part  of 
the  sun's  path  west  of  the  meridian  (Fig.  16).  The  reading  of  the 
foot  of  the  arc  will  give  the  angle  between  the  true  meridian  and 


THE    DIURNAL    MOTION"    OF   THE    SUN 


13 


the  vertical  plane  containing  the  sun  at  that  point  of  its  diurnal 
circle.     This  angle  is  its  true  bearing  and  differs  from  its  magnetic 


5° 


15° 


FIG. 15 


bearing  by  the  declination  of  the  compass,  being  evidently  less  than 
the  magnetic  bearing,  if  the  decimation  is  west  of  north.  It  is  also 
called  the  azimuth  of  the  sun's  vertical  circle,  or,  briefly,  of  the  sun. 


FIG. is 


Formerly  azimuth  was  usually  reckoned  from  north  through  the 
west  or  east,  to  180°  at  the  south  point.  It  is  now  customary  to 
measure  it  from  south  through  west  up  to  360°,  so  that  the  azimuth 


14  LABORATORY   ASTRONOMY 

of  a  body  when  east  of  the  meridian  lies  between  180°  and  360°. 
The  present  method  is  more  convenient  because  the  given  angle 
fixes  the  position  of  the  vertical  circle  without  the  addition  of  the 
letters  E.  and  W.  It  is  worthy  of  notice  that  with  this  notation 
the  azimuth  of  the  sun  as  seen  in  northern  latitudes  outside  of  the 
tropics  always  increases  with  the  time ;  and  indeed  this  is  true  of 
most  of  the  bodies  we  shall  have  occasion  to  observe. 

Now  place  the  altitude  quadrant  so  that  its  foot  is  at  a  point  on 
the  circular  protractor  where  the  reading  is  360°  minus  the  azimuth 
of  the  point  just  measured ;  the  sun  at  this  point  of  its  path  is  just  as 
far  east  of  the  meridian  as  it  was  west  of  the  meridian  at  the  point 
last  considered,  and  it  will  be  found  that  the  altitude  of  the  two 
points  is  the  same.  On  the  path  shown  in  Fig.  16  the  altitude  is 
45°  at  the  points  whose  azimuths  are  60°  and  300°  (60  E.  of  S.). 

This  fact,  that  equal  altitudes  of  the  sun  correspond  to  equal 
azimuths  east  and  west  of  the  true  meridian,  is  an  important  one, 
and  will  presently  be  made  use  of  to  enable  us  to  determine  the 
position  of  the  true  meridian  with  a  greater  degree  of  precision. 

THE   EQUINOCTIAL 

We  shall  find  it  convenient  to  draw  upon  the  hemisphere  another 
line,  which  plays  an  important  role  in  astronomy,  the  great  circle 
90°  from  the  pole.  Placing  the  steel  point  of  the  compasses  at 
the  zenith,  open  the  legs  until  the  pencil  point  just  comes  to  the 
horizon  plane  where  the  spherical  surface  meets  it,  so  that  if  it 
were  revolved  about  the  zenith,  the  pencil  point  would  move  in 
the  horizon.  The  compass  points  now  span  an  arc  of  90°  upon  the 
hemisphere.  Place  the  steel  point  at  the  pole,  and  draw  as  much 
of  a  great  circle  as  can  be  described  on  the  sphere  above  the  horizon. 
This  will  be  just  one-half  of  the  great  circle,  and  will  cut  the  horizon 
in  the  east  and  west  points.  The  new  circle  is  called  the  equinoctial 
or  celestial  equator  (Fig.  17). 

We  have  seen  that  the  path  of  the  sun  over  the  dome  of  the 
heavens  appears  to  be  a  small  circle  described  from  east  to  west 
about  a  fixed  point  in  the  dome  as  a  pole.  The  ancient  explanation 
of  this  fact  was  that  the  sun  is  fixed  in  a  transparent  spherical  shell 


THE  DIURNAL  MOTION  OF  THE  SUN 


15 


of  immense  size  revolving  daily  about  an  axis,  the  earth  being  a 
plane  in  the  center  of  unknown  extent,  but  whose  known  regions 
are  so  small  compared  to  the  shell  that  from  points  even  widely 
separated  on  the  earth  the  appearance  is  the  same ;  just  as  the 


FIG.  17 

apparent  direction  and  motion  of  the  sun  would  be  practically 
the  same  on  our  hemisphere  to  a  microscopic  observer  at  the 
center,  and  to  another  anywhere  within  one-hundredth  of  an  inch 
of  the  center.  When  observations  were  made,  however,  at  points 
some  hundreds  of  miles  apart  on  the  same  meridian,  very  per- 
ceptible differences  were  found,  whose  nature  will  be  understood 
from  a  comparison  of  the  hemisphere  (Fig.  18  a),  plotted  from 


FIG.  18 

observations  made  Aug.  8,  1897,  at  a  point  in  Canada,  not  far 
from  Quebec,  with  a  second  hemisphere  (Fig.  18  &),  on  which  is 
shown  the  path  of  the  sun  on  the  same  date  derived  from  observa- 
tion of  the  shadow  of  a  pin-head  at  Polfos  in  Norway.  It  appears 
on  comparison  that  the  distance  of  the  pole  above  the  north  horizon 


16  LABORATORY    ASTRONOMY 

is  considerably  greater  in  the  latter,  while  the  equator  is  just  as 
much  nearer  the  southern  horizon ;  the  sun  is  at  the  same  distance 
from  the  equator  in  each  case.  This  fact  cannot  be  explained  on 
the  supposition  that  the  horizon  planes  of  the  two  places  are  the 
same,  for  in  that  case  we  should  have  the  spherical  shell  which 
contains  the  sun  revolving  at  the  same  time  about  two  different 
.fixed  axes,  which  is  impossible.  It  is  not,  however,  improbable 
that  the  earth's  surface  should  be  curved,  if  we  can  admit  as 
a  possibility  that  the  direction  of  gravity,  which  is  perpendicular 
to  a  horizontal  plane,  may  be  different  at  different  places.  That 
the  earth's  surface  in  the  east  and  west  direction  is  curved,  we 
know;  for  men  have  traversed  it  from  east  to  west  and  returned 
to  the  starting  point,  so  that  we  have  good  reason  to  believe  that  its 
surface  is  everywhere  curved.  Long  before  this  conclusive  proof 
was  obtained,  however,  the  globular  form  of  the  earth  was  inferred 
on  good  grounds. 

It  was  early  suggested  (regarding  the  fact  that,  if  the  sun  is  fixed 
in  a  shell,  that  shell  is  of  enormous  size  as  compared  with  the  earth) 
that  it  is  inherently  more  probable  that  the  apparent  motion  of 
the  sun  is  due  to  a  rotation  of  the  spherical  earth  about  an  axis 
passing  through  the  earth's  center  and  the  poles  of  the  sun's  circle. 
This  argument  is  greatly  strengthened  when  we  investigate  the 
apparent  motion  of  the  stars  in  connection  with  their  size  and  dis- 
tance, and  it  is  now  beyond  a  doubt  that  this  is  the  true  explanation 
of  the  apparent  diurnal  motion  of  the  sun. 

LATITUDE   EQUALS   ELEVATION   OF    THE   POLE 

This  subject  is  treated  in  all  text-books  on  descriptive  astronomy, 
and  it  is  pointed  out  that  the  pole  of  the  sun's  path  is  the  point 
where  the  line  of  the  earth's  axis  of  rotation  cuts  the  sky,  and  the 
equinoctial  or  celestial  equator  is  the  great  circle  in  which  the  plane 
of  the  earth's  equator  cuts  the  sky.  The  fact  is  proved  also  that 
the  elevation  of  the  pole  above  the  horizon  at  any  place  is  equal  to 
the  latitude  of  the  place. 

This  angle,  as  measured  on  the  hemisphere  shown  in  Fig.  18  a,  is 
47°,  and  on  the  hemisphere  of  Fig.  18  b  is  62°.  The  latitudes  of 


THE  DIURNAL  MOTION  OF  THE  SUN 


17 


Quebec  and  Polfos  as  determined  by  more  accurate  measures  are 
46°  50'  and  61°  57'. 

It  is  easy  to  see  that  the  arc  of  the  meridian  from  the  zenith 
to  the  equinoctial  is  also  equal  to  the  latitude,  while  the  arc  from 
the  south  point  of  the  horizon  to  the  equator  and  that  from  the 
zenith  to  the  pole  are  each  equal  to  90°  minus  the  latitude,  or,  as  it 
is  usually  called,  the  co-latitude. 

It  will  be  well  here,  as  in  all  our  measurements,  to  form  some  idea 
of  the  accuracy  of  our  results.  As  one  degree  on  our  hemisphere 
is  quite  exactly  equal  to  lmm,  a  quantity  easily  measured  by  ordi- 
nary means,  it  is  not  difficult  with  ordinary  care  to  determine  the 


FIG.  19 

pole  of  the  sun's  path  so  closely  that  no  observed  point  lies  more 
than  a  degree  from  the  path.  The  pole  is  then  fixed  within  one 
degree  unless  the  length  of  the  path  is  very  short ;  usually  if  the 
path  is  more  than  90°  in  length  the  pole  may  be  placed  within  less 
than  a  degree  of  its  true  place  and  the  latitude  measured  with  an 
error  of  less  than  one  degree. 


HOUR-ANGLE   OF   THE    SUN 

Open  the  dividers  as  before  (see  p.  14)  so  as  to  draw  a  great  circle. 
Place  the  steel  point  upon  the  place  of  the  sun,  S,  on  its  diurnal 
circle  at  the  time  of  the  last  observation  in  the  afternoon  (Fig.  19), 
and  with  the  pencil  point  strike  a  small  arc  cutting  the  equator  at  Q. 


18  LABORATOKY   ASTRONOMY 

Place  the  steel  point  where  this  arc  cuts  the  equator,  and  draw  a 
great  circle  which  will  pass  through  the  sun's  place  and  the  pole ; 
notice  that  it  also  cuts  the  equator  at  right  angles.  Such  a^  circle  is 
called  an  hour-circle.  It  is  the  intersection  of  the  surface  of  the 
sphere  with  a  plane  that  passes  through  the  poles  and  the  place  of 
the  sun.  The  number  of  degrees  in  the  arc  of  the  equator,  included 
between  the  meridian  and  the  hour-circle  which  passes  through  the 
sun,  is  called  the  hour-angle  of  the  sun.  By  the  ordinary  convention 
of  solid  geometry  it  measures  the  wedge  angle  between  the  plane  of 
the  hour-circle  and  the  plane  of  the  meridian.  If  a  book  be  placed 
with  its  back  in  the  line  from  the  pole  to  the  center  of  the  sphere, 
and  with  its  title-page  to  the  west,  and  the  western  cover  opened 
till  it  is  in  the  plane  of  the  hour-circle,  while  the  title-page  is  in 
the  plane  of  the  meridian,  the  wedge  angle  between  the  title-page 
and  the  cover  will  be  the  hour-angle  and  will  be  measured  by  the 
arc  of  the  equator  indicated  above.  It  is  reckoned  as  increasing 
from  the  meridian  towards  the  west  in  the  direction  in  which  the 
cover  is  opened.  If  the  hour-circle  of  the  first  morning  observa- 
tion is  determined  in  the  same  way,  the  hour-angle  measured  in 
the  opposite  direction  from  the  meridian  is  sometimes  called  the 
hour-angle  east  of  the  meridian ;  but  more  commonly  by  astronomers 
this  value  is  subtracted  from  360°,  and  the  angle  thus  obtained  is 
called  the  hour-angle,  this  being  more  convenient  because  the  hour- 
angle  of  the  sun  thus  measured  constantly  increases  with  the  time 
as  the  sun  pursues  its  course ;  being  0°  at  noon,  180°  at  midnight, 
360°  at  the  next  noon,  etc. 

UNIFORM  INCREASE   OF   HOUR-ANGLE 

Let  us  now  examine  more  carefully  the  truth  of  the  surmise  pre- 
viously made,  that  the  arc  of  the  sun's  path  between  two  successive 
observations  is  proportional  to  the  interval  of  time  between  the 
observations.  Draw  the  hour-circles  of  the  sun  at  each  point  of 
observation  (Fig.  20)  ;  measure  the  arc  on  the  equator  between  the 
first  and  the  last  hour-circles ;  divide  by  the  number  of  minutes 
between  the  two  times.  This  will  give  the  average  increase  of 
hour-angle  per  minute.  Multiply  this  increase  by  the  difference  in 


THE  DIURNAL  MOTION  OF  THE  SUN 


19 


minutes  of  each  of  the  observed  times  from  the  time  of  the  first 
observation,  and  compare  with  the  progressive  increase  of  the  hour- 
angle  as  measured  off  on  the  equator  by  means  of  the  graduated 
quadrant.  They  will  be  found  to  be  nearly  the  same  in  each  case. 
It  is  thus  shown  that  the  hour-angle  of  the  sun  increases  uniformly 
with  the  time.  The  rate  is  nearly  a  quarter  of  a  degree  per  minute, 
since  360°  are  described  in  24  hours.  Notice  that  when  the  hour- 
angle  is  zero,  the  actual  time  by  the  watch  is  not  very  far  from  12 
o'clock  (in  extreme  cases  it  may  be  45  minutes,  if  the  clock  is  keep- 
ing standard  time),  and  that  if  the  hour-angle  in  degrees  (west  of 
the  meridian)  is  divided  by  15,  the  number  of  hours  differs  from  the 


FIG.  20 

watch  time  just  as  much  as  the  time  of  meridian  passage  differs 
from  12  hours.  In  fact,  the  hour-angle  of  the  sun  measures  what 
is  called  apparent  solar  time,  i.e.,  when  H.A.  =  15°,  it  is  1  o'clock; 
H. A.  =  75°,  it  is  5  o'clock  j  H.A.  =  150°,  10  o'clock,  etc. ;  those  angles 
east  of  the  meridian  lying  between  180°  and  360°,  i.e.,  between  12h 
and  24h,  so  that  12  hours  must  be  subtracted  to  give  the  correct  hours 
by  the  ordinary  clock,  which  divides  the  day  into  two  periods  of  24 
hours  each ;  for  instance,  if  H.A.  =  270°,  it  is  18h  past  noon  or  6  A.M. 
of  the  next  day.  Astronomical  clocks  usually  show  the  hours  con- 
tinuously from  0  to  24,  thus  avoiding  the  necessity  of  using  A.M. 
and  P.M.  to  discriminate  the  period  from  noon  to  midnight  and  from 
midnight  to  noon. 


20  LABORATORY    ASTRONOMY 


DECLINATION   OF   THE    SUN 

The  distance  of  the  sun's  path  from  the  celestial  equator,  meas- 
ured along  the  arc  of  an  hour-circle,  is  called  its  declination,  and 
will  be  found  appreciably  the  same  at  all  points.  It  requires  more 
delicate  observation  than  ours  to  find  that  it  changes  during  the 
few  hours  covered  by  our  observation.  If,  however,  the  observa- 
tion be  repeated  after  an  interval,  say,  of  two  weeks  at  any  time 
except  for  a  month  before  or  after  the  20th  of  June  or  December, 
it  will  be  found  that  although  the  sun  at  the  second  observation 
describes  a  circle,  this  circle  is  not  in  the  same  position  with  regard 
to  the  equator  —  that  its  declination  has  changed  (between  March 
13  and  27,  for  instance,  by  about  5°.5).  The  inference  to  be  drawn 
is  that  even  during  the  period  of  our  observation  the  sun's  path  is 
not  exactly  parallel  to  the  equator,  although  our  observations  are 
not  delicate  enough  to  show  that  fact. 

It  is  true  in  general,  as  in  this  case,  that  the  first  rude  meas- 
urements applied  to  the  heavenly  bodies  give  results  which  when 
tested  by  those  covering  a  longer  time,  or  made  with  more  delicate 
instruments,  are  found  to  require  correction. 


CHAPTER   II 
THE  MOON'S   PATH  AMONG  THE  STARS 

NEXT  to  the  diurnal  motion  of  the  sun  the  most  conspicuous 
phenomenon  is  the  similar  motion  of  the  stars  and  the  moon  j  this 
will  form  the  subject  of  a  future  chapter. 

The  study  of  the  moon,  however,  discloses  a  new  and  interesting 
motion  of  that  body.  It  partakes  indeed  of  the  daily  motion  of 
the  heavenly  bodies  from  east  to  west,  but  it  moves  less  rapidly, 
requiring  nearly  25  hours  to  complete  its  circuit  instead  of  24,  as 
do  the  sun  and  stars,  and  returning  to  the  meridian  therefore 
about  an  hour  later  on  each  successive  night. 

In  consequence  of  this  motion  it  continually  changes  its  place 
with  reference  to  the  stars,  moving  toward  the  east  among  them 
so  rapidly  that  the  observation  of  a  few  hours  is  sufficient  to  show 
the  fact.  At  the  same  time  its  declination  changes  like  that  of  the 
sun,  but  much  more  rapidly. 

We  should  begin  early  to  study  this  motion,  and  it  will  be  found 
interesting  to  continue  it  at  least  for  some  months  at  the  same  time 
that  other  observations  are  in  progress  —  a  very  few  minutes  each 
evening  will  give  in  the  course  of  time  valuable  results. 

POSITION   BY   ALIGNMENT   WITH   STARS 

The  first  method  to  be  used  consists  in  noting  the  moon's  place 
with  reference  to  neighboring  stars  at  different  times.  Some  sort 
of  star  map  is  necessary  upon  which  the  places  of  the  moon  may  be 
laid  down  so  that  its  path  among  the  stars  may  be  studied.  As 
the  configurations  that  offer  themselves  at  different  times  are  of 
great  variety,  it  will  be  well  to  give  a  few  examples  of  actual 
observations  of  the  moon's  place  by  this  method. 

Dec.  12,  1899,  at  X12h  Om  P.M.,  the  moon  was  seen  to  be  near 
three  unknown  stars,  making  with  them  the  following  configuration, 

21 


22  LABORATORY   ASTRONOMY 

which  was  noted  on  a  slip  of  paper  as  shown  in  Fig.   21.     The 

relative  size  of  the  stars  is  indicated  by  the  size  of  the  dots.     (The 

original  papers  on  which  the  observations  are  made  should  be  care- 

ms  -Dec  12*  fr^ty  preserved ;   indeed,  this  should  always  be  the 

practice  in  all  observations.) 

At  the  same  time,  for  purposes  of  identification, 
it  was  noted  that  the  group  of  stars  formed,  with 
/  Capella  and  the  brightest  star  in  Orion,  both  of 

A  a  mmeiricatfi  urt   wn^cn  were  known  to  the  observer,  a  nearly  equi- 
lateral triangle.     It  was  also  noted  that  the  moon 
was   about   6°  from  the  farthest  star,  this  being 
estimated  by   comparison  with  the  known    distance   between  the 
"pointers"  in  the  "Dipper"  (about  5°).     With  these  data  it  was 
easily  found  by  the  map  that  these  stars  were  the  brightest  stars 
in  Aries,  and  the  moon  was  plotted  in  its  proper  place  on  the  map 
(page  24). 

December  13,  at  5h  35m  P.M.,  the  moon  was  £°  (half  its  diameter) 
below  (south  of)  a  line  drawn  from  Aldebaran  (identified  by  its 
position  with  reference  to  Capella  and  Orion  and  by  the  letter  V  of 
stars  in  which  it  lies,  the  Hyades)  to  the  faintest  of  the  three 
reference  stars  of  December  12.  It  was  also  about  f°  west  of  a 
line  between  two  unknown  stars  identified  later  as  Algol  (equi- 
distant from  Capella  and  Aldebaran)  and  y  Ceti  (at  first  supposed 
on  reference  to  the  map  to  be  a  Ceti, 

Dec.  13d5*3Sm        <<w  0^ 

but  afterward  correctly  identified  by  • 

comparing  the  map  with  the  heavens).  \ 

The  original  observation  is  given 
below  (Fig.  22)  of  about  one-half  the 
size  of  the  drawing,  all  except  the 
underscored  names  being  in  pencil. 
The  underscored  names  are  in  ink  and  «.  \ 

made  after  the  stars  were  identified. 
This  is  a  useful  practice  when  addi- 
tions are  made  to  an  original,  so  that  subsequent  work  may  not  be 
given  the  appearance  of  notes  made  at  the  time  of  observation.     It 
is  well  to  give  on  the  sketch  map  several  stars  in  the  neighborhood 
of  those  used  for  alignment,  to  facilitate  identification. 


THE    MOON'S    PATH   AMONG   THE    STARS  23 

The  alignment  was  tested  by  holding  a  straight  stick  at  arm's 
length  parallel  to  the  line  joining  the  stars. 

December  14,  6h  30m  P.M.  Moon  on  a  line  from  Algol  through 
the  Pleiades  (known)  about  2^°  (5  diameters  of  moon)  beyond  the 
latter,  which  were  very  faint  in  ag^^^  .  *c«y»««a  sec  ^ds^o^ 

the  strong  moonlight.     No  figure.  \  v 

December  15, 5h  10m  P.M.     Moon  \         \ 

in  a  line  between  Capella  and  Aide-  \       \ 

baran.    Line  from  Pleiades  to  moon         -_.    .  .     \      • 

filaun     •  -,          \ 

bisects  line  from  Aldebaran  to  (3  \    \ 

Tauri    (identified    by    relation    to 
Aldebaran  and  Capella). 

9h  25m  P.M.     Moon  in  line  from 

3  Aurigse  to  Aldebaran  (Fig.  23). 

FIG.  23 

(NOTE.  — Henceforth  details  of  identification  are  omitted.) 

December  16,  7h  40m  P.M.     Moon  almost  totally  eclipsed  2J°  east 

of  line  from  ft  Aurigae  to  y  Orionis ;  same  distance  from,  ft  Tauri  as 

£  Tauri  (revised  estimate  about  £°  nearer  ft  Tauri 

.\  December    18,    10h    30m    P.M.       Observation 

\  snatched  between  clouds.     Moon's  western  edge 

tangent  to  line  from  a  Geminorum  to  Procyon 
and  about  1°  north  of  center  of  that  line. 
0  \  In  the  sketch  maps  above  no  great  accuracy  is 

attempted  in  placing  the  stars,  but  in  the  final 
•  "•        plotting  on  the  map  the  directions  of  the  notes 

\  are  carefully  followed.     The  plotting  should  be 

\y  done  as   soon  as  possible  after  the  observation 

is  made,  for  even  a  hasty  comparison  with  the 
map  will  often  show  that  stars  have  been  mis- 
identified  or  that  there  is  some  obvious  error  in 
FlG  24  the  notes,  which  may  be  rectified  at  once  if  there 

is  an  opportunity  to  repeat  the  observation.  Such 
a  case  occurs  in  the  observations  of  December  13  recorded  above, 
where  y  Ceti  was  mistaken  for  a. 


24 


LABORATORY    ASTRONOMY 


PLOTTING  POSITIONS  OF  THE  MOON  ON  A  STAR  MAP 

Figure  25  shows  the  positions  of  the  moon  plotted  from  the  fore- 
going observations,  together  with  the  lines  of  construction  from 
which  they  were  determined. 

A  drawing  should  be  made  of  the  shape  of  the  illuminated  portion 
of  the  moon  at  each  observation,  and  the  direction  among  the  stars 


FIG.  25 


of  the  line  joining  the  points  of  the  horns  (cusps)  for  future  study 
of  the  cause  of  the  moon's  changes  of  phase. 

If  the  star  map  accompanying  this  book  is  used,  the  identification 
of  the  stars  consists. in  determining  which  of  the  dots  represents 
the  star  of  reference ;  the  name  may  be  determined  by  reference 
to  the  list;  thus  the  two  stars  near  the  line  XXIV  on  the  upper 
portion  of  the  map  are  "a  Andromedee  Oh  5m  +  29°"  and  "yPegasi 
Qh  gm  _^_  140  »  rpj^  meanjng  which  attaches  to  these  numbers  is 
given  in  Chapter  IV.  It  is  a  good  plan  to  keep  a  copy  of  the 
map  on  which  to  note  the  names  for  reference  as  the  stars  are 
learned ;  most  of  the  conspicuous  ones  will  soon  be  remembered 
as  they  are  used. 


THE    MOON  S    PATH   AMONG    THE    STARS 


25 


THE   MOON'S   PLACE   FIXED    BY  ITS   DISTANCE   FROM 
NEIGHBORING   STARS 

One  month's  observation  by  this  method  will  show  that  the  moon's 
path  is  at  all  points  near  to  the  curved  line  drawn  on  the  map, 
which  is  called  the  ecliptic  and  which  is  explained  on  page  70. 
To  establish  more  accurately  its  relations  to  this  line  it  will  be 
advisable  in  the  later  months  to  adopt  a  more  accurate  means  of 
observation,  although  when  the  moon  is  very  near  a  bright  star,  its 
position  may  be  quite  accurately  fixed  by  the  means  that  we  have 
indicated ;  and  if  it  chances  to  pass  in  front  of  a  bright  star  and 
produce  an  occupation,  the  moon's  position  is  very  accurately  fixed 
indeed,  as  accurately  as  by  any  method.  But  such  opportunities 
are  rare,  and  for  continuous  accurate  observation  we  should  have  a 
means  of  measuring  the  distance  of  the  moon  from  stars  that  are 
at  a  considerable  distance  from  it.  An  instrument  sufficiently  accu- 
rate for  our  purpose  is  the  cross-staff  described  below.  It  should  be 
mentioned  that,  on  ac- 
count of  the  distortion 
of  the  map,  the  place  of 
the  moon  is  usually  more 
accurately  given  by  dis- 
tances from  the  com- 
parison stars  than  by 
alignment.  The  sextant 
may  be  used  instead  of 
the  cross-staff,  but  is 
less  convenient  and  also 
more  accurate  than  is 
necessary. 

The  Cross-staff.  —  The 
cross-staff  (Fig.  26)  con- 
sists of  a  straight  graduated  rod  upon  which  slides  a  "  transversal " 
or  "  cross  "  perpendicular  to  the  rod ;  one  end  of  the  staff  is  placed 
at  the  eye  and  the  "  cross  "  is  moved  to  such  a  place  that  it  just 
fills  the  angle  from  one  object  to  another ;  its  length  is  then  the 
chord  of  an  arc  equal  to  the  angle  between  the  objects  as  seen  from 


FIG.  26 


26  LABORATOKY    ASTRONOMY 

that  end  of  the  staff  at  which  the  eye  is  placed.  The  figure,  which 
is  taken  from  an  old  book  on  navigation,  illustrates  the  use  of  this 
instrument  for  measuring  the  sun's  altitude  above  the  sea  horizon ; 
the  rod  in  the  position  shown  indicates  that  the  sun's  altitude  is 
about  40°. 

Obviously  a  given  position  of  the  cross  corresponds  to  a  definite 
angle  at  the  end  of  the  rod,  and  the  rod  may  be  graduated  to  give 
this  angle  directly  by  inspection,  or  a  table  may  be  constructed  by 
which  the  angle  corresponding  to  any  division  of  the  rod  may  be 
found  ;  such  a  table  is  given  on  page  27.  For  our  purpose  an  instru- 
ment of  convenient  dimensions  is  made  by  using  a  cross  20  cm.  in 
length,  sliding  on  a  rod  divided  into  millimeters  (Fig.  27)  ;  this  may 
be  used  for  measuring  angles  up  to  30°,  which  is  enough  for  our 


l.i.;.i.r.M.U.I*M 


FIG.  27 

purpose.  The  smallest  angle  that  can  be  measured  is  about  12°, 
which  corresponds  to  a  chord  of  £  of  the  radius ;  but  by  making  a 
part  of  the  cross  only  10  cm.  long,  as  shown  in  the  figure,  we  may 
measure  angles  from  6°  upwards,  and  for  smaller  angles  may  use 
the  thickness  of  the  cross,  which  is  5  cm.,  and  thus  measure  angles 
as  small  as  3° ;  the  longer  cross  will  not  give  good  results  above 
30°,  as  a  slight  variation  of  the  eye  from  the  exact  end  of  the  rod 
makes  a  perceptible  difference  in  the  value  of  the  angles  greater 
than  30°. 

Measures  with  the  Cross-staff.  —  As  an  example  of  the  use  of  the 
cross-staff,  the  following  observations  are  given:  They  were  made 
with  a  staff  about  3  feet  in  length,  graduated  by  marking  the  point 
for  each  degree  at  the  proper  distance  in  millimeters  from  the  eye 
end  of  the  staff,  as  given  by  Table  II  on  page  27.  After  the  points 
were  marked  a  straight  line  was  drawn  through  each  entirely  across 
the  rod,  using  the  cross  itself  as  a  ruler ;  graduations  were  thus 
made  on  one  side  for  use  with  the  20  cm.  cross,  on  the  other  for  the 


THE    MOON  S    PATH    AMONG    THE    STARS 


27 


TABLE  I  —  ANGLE  SUBTENDED  BY  CROSSES 

TABLE  II 

Distance 
from 
Eye 

LEXGTH  OF  CROSS 

Distance 
from 
Eye 

LENGTH  OF  CROSS 

Angle 
subtended  by 
20  cm.  Cross 

20cm. 

10  cm. 

5  cm. 

20cm. 

10  cm. 

5  cm. 

lOQcm 

11°.  4 

5°.  7 

2°.  9 

62cm 

18°.3 

9°.  2 

4°.  6 

99 

11  .5 

5  .8 

2  .9 

61 

18.6 

9.4 

4  .7 

98 

11  .6 

5.8 

2  .9 

60 

18  .9 

9.5 

4  .8 

97 

11  .8 

5  .9 

3.0 

59 

19  .2 

9  .7 

4  .9 

96 

11  .9 

6.0 

3.0 

58 

19.6 

9.9 

4.9 

95 

12.0 

6.0 

3.0 

57 

19.9 

10  .0 

5  .0 

94 

12  .1 

6  .1 

3  .0 

56 

20.2 

10  .2 

5  .0 

12° 

951mm 

93 

12  .3 

6.2 

3.1 

55 

20  .6 

10  .4 

5  .2 

13 

878 

92 

12  .4 

6  .2 

3  .1 

54 

21  .0 

10  .6 

5  .3 

14 

814 

91 

12  .5 

6  .3 

3.1 

53 

21  .4 

10  .8 

5  .4 

15 

760 

90 

12  .7 

6.4 

3.2 

52 

21  .8 

11  .0 

5  .5 

16 

711 

89 

12.8 

6  .4 

3  .2 

51 

22  .2 

11  .2 

5  .6 

17 

669 

88 

13  .0 

6  .5 

3.3 

50 

22  .6 

11  .4 

5  .7 

18 

631 

87 

13  .1 

6.6 

3  .3 

49 

23  .1 

11  .6 

5  .8 

19 

598 

86 

13.3 

6  .7 

3  .3 

48 

23  .5 

11  .9 

6.0 

20 

567 

85 

13  .4 

6  .7 

3.4 

47 

24.0 

12  .1 

6.1 

21 

540 

84 

13.6 

6  .8 

3  .4 

46 

24.5 

12  .4 

6  .2 

22 

514 

83 

13.7 

6  .9 

3.5 

45 

25  .1 

12  .7 

6  .4 

23 

491 

82 

13  .9 

7.0 

3.5 

44 

25.6 

13  .0 

6  .5 

24 

470 

81 

14.1 

7  .1 

3  .5 

43 

26  .2 

13.3 

6  .7 

25 

451 

80 

14  .3 

7  .2 

3.6 

42 

26  .8 

13  .6 

6.8 

26 

433 

79 

14  .4 

7.2 

3  .6 

41 

27  .4 

13  .9 

7  .0 

27 

416 

78 

14  .6 

.7  .3 

3  .7 

40 

28  .1 

14.3 

7  .2 

28 

401 

77 

14  .8 

7  .4 

3.7 

39 

28.8 

14  .6 

7  .3 

29 

387 

76 

15  .0 

7  .5 

3  .8 

38 

29.5 

15  .0 

7  .5 

30 

373 

75 

15  .2 

7  .6 

3.8 

37 

30.2 

15  .4 

7  .7 

31 

361 

74 

15  .4 

7  .7 

3  .9 

36 

31  .0 

15.8 

7  .9 

32 

349 

73 

15  .6 

7  .8 

3  .9 

35 

31  .9 

16.3 

8  .2 

33 

338 

72 

15  .8 

7  .9 

4  .0 

34 

32.8 

16  .7 

8.4 

34 

327 

71 

,16  .0 

8  .1 

4  .0 

33 

33.7 

17  .2 

8.7 

35 

317 

70 

16  .3 

8  .2 

4  .1 

32 

34.7 

17.7 

8  .9 

36 

308 

69 

16  .5 

8.3 

4  .2 

31 

35  .8 

18.3 

9.2 

37 

299 

68 

16  .7 

8  .4 

4  .2 

30 

36  .9 

18  .9 

9  .5 

38 

290 

67 

17  .0 

8  .5 

4.3 

29 

38.1 

19.6 

9  .9 

39 

282 

66 

17.2 

8.7 

4  .3 

28 

39.3 

20  .2 

10  .2 

40 

275 

65 

17  .5 

8  .8 

4  .4 

27 

40.6 

21  .0 

10.6 

64 

17  .7 

8  .9 

4.5 

26 

42  .1 

21  .8 

11  .0 

63 

18  .0 

9.1 

4  .5 

25 

43  .6 

22  .6 

11  .4 

28  LABORATORY   ASTRONOMY 

10  cm.  cross,  and  on  one  edge  for  the  thickness  of  the  cross.     By 
means  of  these  graduations  the  angle  subtended  by  the  cross  in  any 
position  is  read  directly  from  the  scale,  quarters  or  thirds  of  a 
degree  being  estimated  and  recorded  in  minutes  of  arc. 
The  observations  are : 

1900.     January  2.     5h  15m. 

Moon  to  e  Pegasi,  35°  45' 

"       "      Altair,  26   30 

"       "      Fomalhaut,  41    40 

January  3.     6h  Om. 

Moon  to  e  Pegasi,  23°  30' 

"       "      Altair,  29   20 

"       "  j8  Aquarii,  8   20 

January  4.     5h  20m. 

Moon  to  e  Pegasi,  17°  40' 

"       "  |8  Aquarii,  8   30 

"       "5  Capricorni,  9   45 

January  6.     5h  50m. 

Moon  to  7  Pegasi,  12°    0' 

"       "  a  Pegasi,  16   40 

"      "  e  Pegasi,  33   30 

January  7.     5h  45m. 

Moon  to  7  Pegasi,  9°  40' 

"      "  j8  Arietis,  19   45 

"       "a  Andromedee,  21    15 
"       "  j8  Ceti,  27   30 

January  8.     6h  Om. 

Moon  to  a  Arietis,  11°    0' 

•"       "  7  Pegasi,  21    30 

January  9.     10h  Om. 

Moon  to  a  Arietis,  9°  45' 

"       "     Alcyone,  16     0 

"      "  a  Ceti,  15   30 

To  represent  these  observations  on  the  star  map,  open  the  com- 
passes until  the  distance  of  the  pencil  point  from  the  steel  point  is 
equal  to  the  measured  distance  —  making  use  for  this  purpose  of 
the  scale  of  degrees  in  the  margin,  and  then  with  the  steel  point 


THE    MOON'S    PATH    AMONG    THE    STARS 


29 


carefully  centered  on  the  comparison  star,  strike  a  short  arc  with 
the  pencil  point  near  the  estimated  position  of  the  moon ;  the  inter- 
section of  any  two  of  these  arcs  fixes  the  position  of  the  moon.  If 
the  different  stars  give  different  points,  those  nearest  the  moon  may 


JO- 


Pleiades 


-20- 


tyades 


-w- 


FlQ.  28 

be  assumed  to  give  results  nearer  the  truth.  Fig.  28  shows  the 
positions  of  the  moon  January  6  to  January  9  as  plotted  from  the 
above  measures. 

Length  of  the  Month.  —  If  it  happens  that  one  of  the  positions  ob- 
served in  the  second  month  falls  between  the  places  obtained  on  two 
successive  days  of  the  first  month,  or  vice  versa,  a  determination  of 
the  moon's  sidereal  period  may  be  made  by  interpolation.  Thus,  on 
plotting  the  observation  of  December  12  (p.  22),  which  places  the 
moon  between  the  two  observations  on  January  8d  6h  Om  and  Janu- 
ary 9d  10h  Om,  its  distance  from  the  former  is  6°.0  and  from  the  latter 
10°.0,  while  the  interval  is  28h ;  the  moon's  place  on  December  12  at 

12h  Om  is  therefore  the  same  as  on  January  8  at  6h  +  —  X  28h,  or  Jan- 
uary 8d  16h.5,  that  is,  January  9  at  4h  30m  A.M.,  and  the  interval 
between  these  two  times  is  27d  4h  30m,  which  is  the  time  required  for 
the  moon  to  make  a  complete  circuit  among  the  stars  or  the  length 


30  LABORATORY    ASTRONOMY 

of  the  sidereal  month.  This  is  a  fairly  close  approximation ;  the 
observation  of  December  12  having  been  made  under  favorable 
circumstances,  the  configuration  being  well  denned  and  the  stars 
near,  so  that  the  position  on  that  date  by  alignment  is  nearly  as 
accurate  as  those  determined  by  the  measures  on  January  8  and  9. 
After  three  months  the  moon  comes  nearly  to  the  same  position  at 
about  the  same  time  in  the  evening,  so  that  it  is  convenient  to  deter- 
mine its  period  without  interpolation  by  observing  the  time  when  the 
moon  comes  into  the  same  star  line  as  at  the  previous  observation ; 
moreover,  the  interval  being  three  months,  an  error  of  an  hour  in 
the  observed  interval  causes  an  error  of  only  20m  in  the  length  of 
the  month. 

THE   MOON'S   NODE 

When  a  sufficiently  large  number  of  observations  have  been  plot- 
ted to  give  a  general  idea  of  the  moon's  path  among  the  stars,  a 
smooth  curve  is  to  be  drawn  as  nearly  as  possible  through  all  the 
points  and  this  curve  should  be  compared  with  the  ecliptic,  as  shown 
on  the  map.  Its  greatest  distance  from  the  ecliptic  and  the  place 
where  it  crosses  the  ecliptic  —  the  position  of  the  node  —  should  be 
estimated  with  all  possible  precision.  For  this  purpose,  only  the 
more  accurate  positions  obtained  by  the  cross-staff  should  be  used. 

After  a  few  observations  of  alignment  are  made,  the  student  will 
desire  to  use  the  more  accurate  method  at  once,  but  it  is  better  to 
have  at  least  one  month's  observation  by  the  first  method  (even  if 
the  cross-staff  is  also  used)  for  comparison  with  later  observations  by 
alignment  for  the  purpose  of  determining  the  length  of  the  month, 
as  suggested  above,  without  any  instrumental  aid  whatever. 

The  records  of  the  positions  of  the  node  should  be  preserved  by 
the  teacher  for  comparison  from  year  to  year  to  show  the  motion 
of  this  point  along  the  ecliptic.  The  node,  as  determined  by  the 
observations  above  given,  was  nearly  at  the  point  where  the  ecliptic 
crosses  the  line  from  y  Orionis  to  Capella.  Observations  made  in 
November,  1897,  by  the  method  of  Chapter  IV,  gave  its  place  on  the 
ecliptic  at  a  point  where  the  latter  intersects  a  line  drawn  through 
Castor  and  Pollux,  thus  indicating  a  motion  of  about  40°  in  the 
interval 


THE    MOON'S    PATH    AMONG   THE    STARS  31 

Observations  made  with  the  cross-staff  are  sufficiently  accurate 
to  show  that  the  motion  of  the  moon  is  not  uniform,  but  as  the  dis- 
tortion of  the  map  complicates  the  treatment  of  this  subject,  we 
shall  defer  its  consideration  until  the  method  of  Chapter  V  has 
been  introduced. 

It  will  be  well,  however,  as  soon  as  measures  with  the  cross-staff 
are  begun,  to  devote  a  few  minutes  each  evening  to  measures  of  the 
moon's  diameter  with  an  instrument  measuring  to  10",  such  as  a  good 
sextant ;  or,  better,  a  telescope  provided  with  a  micrometer,  in  order 
to  show  the  variations  of  the  moon's  apparent  size  at  different  parts 
of  its  orbit.  The  relative  distances  of  the  moon  from  the  earth  as 
inferred  from  these  measures  should  be  compared  with  the  varia- 
tions of  her  angular  motion  as  read  off  from  the  chart ;  although 
on  account  of  the  distortion  referred  to  above,  it  will  not  be  possible 
to  show  more  than  the  fact  that  when  the  moon  is  nearest,  her 
angular  motion  about  the  earth  is  greatest,  and  vice  versa. 

The  sextant  or  micrometer  may  henceforward  be  used  also  for 
observations  of  the  sun's  diameter,  which  should  be  measured  as 
often  as  once  a  week  for  a  considerable  period. 

When  the  moon's  diameter  is  measured,  a  rough  estimate  of  her 
altitude  should  be  made  in  order  to  make  the  correction  for  aug- 
mentation in  a  future  more  accurate  discussion  of  the  measures  for 
determining  the  eccentricity  of  her  orbit. 

DETERMINING   THE    ERRORS    OF    THE    CROSS-STAFF 

Observations  with  the  cross-staff  are  most  easily  made  just  before 
the  end  of  twilight  or  in  full  moonlight,  so  that  the  cross  may  be 
seen  dark  against  a  dimly  lighted  background.  When  used  for 
measuring*  the  distance  of  stars  in  full  darkness,  it  is  convenient 
to  have  a  light  so  placed  behind  the  observer  that,  while  invisible 
to  him,  it  shall  dimly  illuminate  the  arms  of  the  cross. 

As  the  angles  which  are  determined  by  the-  cross-staff,  especially  if 
large,  are  affected  by  the  observer's  habit  of  placing  the  eye  too  near 
to  or  too  far  from  the  end  of  the  staff,  it  is  a  good  plan  to  measure 
certain  known  distances  and  thus  determine  a  set  of  corrections  to 
be  applied,  if  necessary,  to  all  measures  made  with  that  instrument. 


32 


LABORATORY    ASTRONOMY 


The  following  table  gives  the  distances  between  certain  stars  always 
conveniently  placed  for  observation  in  the  United  States,  together 
with  the  results  of  measures  made  upon  them  with  a  cross-staff 
held  in  the  hands  without  support,  and  indicates  fairly  the  accuracy 
which  may  be  obtained  with  this  instrument.  The  back  of  the 
observer  was  toward  the  window  of  a  well-lighted  room,  and  the 
cross  was  plainly  visible  by  this  illumination. 


STARS 

TRUE 
DIS- 
TANCE 

MEASURED  DIS- 
TANCES 

MEAN 

CORREC- 
TION 

aUrsse 

Majoris  to  /3  Ursse  Majoris 

5°.  4 

5°.8  —        — 

5°.  8 

-  0°.4 

a     " 

i  i                     il      y                  i   .                                  it 

10  .0 

10  .5  10°.  6  10°.  6 

10  .6 

-0  .6 

a     " 

it         a  f         u               u 

15  .2 

15  .6  15  .5  15  .7 

15  .6 

-0  .4 

/3     « 

"         "f        " 

19  .9 

20  .0  20  .3  20  .2 

20  .2 

-0  .3 

a     " 

;  t          it  „         a               a 
'1 

25  .7 

26  .6  26  .0  26  .1 

26  .2 

-0  .5 

a      " 

"       "     Polaris 

28  .5 

29  .0  29  .2  28  .9 

29  .0 

-0  .5 

ft     " 

u         u            u 

33  .9 

35  .1  34  .7  34  .4 

34  .7 

-0  .8 

•n    " 

u          u             «  t 

41  .2 

42  .2  42  .0  42  .0 

42  .1 

-  0  .9 

The  measured  distances  are  about  one-half  degree  too  large,  and 
if  a  correction  of  this  amount  is  applied  to  all  angles  measured  by 
this  instrument  up  to  30°,  the  corrected  values  will  seldom  be  so 
much  as  half  a  degree  in  error,  and  the  mean  of  three  readings  will 
probably  be  correct  within  a  quarter  of  a  degree. 


CHAPTER   III 
THE  DIURNAL  MOTION  OF  THE  STARS 

As  the  observations  of  the  moon  require  but  a  few  minutes  each 
evening,  observations  may  be  made  on  the  same  nights  upon  the 
stars.  The  first  object  is  to  obtain  the  diurnal  paths  of  some  of  the 
brighter  stars,  and  as  they  cast  no  shadow  we  must  have  recourse 
to  a  new  method  of  observation  to  determine  their  positions  in  the 
sky  at  hourly  intervals. 

A  simple  apparatus  for  this  purpose  is  represented  in  Fig.  29. 
A  paper  circle  is  fastened  to  the  leveling  board  used  in  the  sun 


FIG.  29 


observations  so  that  the  zero  of  its  graduation  lies  as  nearly  as 
possible  in  the  meridian,  and  a  pin  with  its  head  removed  is  placed 
upright  through  the  center  of  the  circle. 

A  carefully  squared  rectangular  block  about  10  inches  by  8  inches 
by  2  inches  is  placed  against  the  pin  so  that  the  angle  which  its 
face  makes  with  the  meridian  may  be  read  off  upon  the  horizontal 

33 


34 


LABORATORY    ASTRONOMY 


circle.  A  second  paper  circle  is  attached  to  the  face  of  the  block 
with  the  zero  of  its  graduations  parallel  to  the  lower  edge ;  a  light 
ruler  is  fastened  to  the  block  by  a  pin  through  the  center  of  its 
circle  j  the  ruler  may  be  pointed  at  any  star  by  moving  the  block 
about  a  vertical  axis  till  its  plane  passes  through  the  star,  and  then 
moving  the  ruler  in  the  vertical  plane  till  it  points  at  the  star ;  a 
lantern  is  necessary  for  reading  the  circles  and  for  illumination  of 
the  block  and  ruler  in  full  darkness ;  it  should  be  so  shaded  that 
its  direct  light  may  not  fall  on  the  observer's  eye.  Sights  attached 
to  the  ruler  make  the  observation  slightly  more  accurate,  but  also 
rather  more  difficult,  and  without  them  the  ruler  may  be  pointed 
within  half  a  degree,  which  is  about  as  closely  as  the  angles  can  be 
determined  by  the  circles. 


THE   ALTAZIMUTH 

An  inexpensive  form  of  instrument  for  measuring  altitude  and 
azimuth  is  shown  in  Fig.  30.  Here  the  ruler  provided  with 
sights  A,  B  is  movable  about  d,  the  center  of  the  semicircle  E. 

This  semicircle  is  movable  about  an  axis 
perpendicular  to  the  horizontal  circle  F, 
and  its  position  on  that  circle  is  read 
off  by  the  pointer  g,  which  reads  zero 
when  the  plane  of  E  is  in  the  meridian. 
The  circle  F  is  mounted  on  a  tripod 
provided  with  leveling  screws.  If  the 
circle  is  so  placed  that  the  pointer  reads 
zero  when  the  sight-bar  is  in  the  mag- 
netic meridian,  then  its  reading  when 
the  sights  are  pointed  at  any  star  will 
give  the  magnetic  bearing  of  the  star. 
It  will,  however,  be  more  convenient  to 
adjust  the  instrument  so  that  the  pointer 
reads  zero  when  the  sight-bar  is  in  the 
true  meridian. 

To  insure  the  verticality  of  the  standard  a  level  is  attached  to 
the  sight-bar,  and  by  the  leveling  screws  the  instrument  must  be 


FIG.  30 


THE    DIURNAL    MOTION    OF    THE    STARS 


35 


adjusted  so  that  the  circle  E  may  be  revolved  without  causing  the 
level  bubble  to  move.  (See  page  36.) 

A  more  convenient  and  not  very  expensive  instrument  is  the 
altazimuth  or  universal  instrument  shown  in  Fig.  31,  which  contains 
some  additional  parts  by  the  use  of  which  it  may  be  converted  into 
an  equatorial  instrument. 
(See  page  45.)  It  consists 
of  a  horizontal  plate  carry- 
ing a  pointer  and  revolving 
on  an  upright  axis  which 
passes  through  the  center 
of  a  horizontal  circle  grad- 
uated continuously  from  0° 
to  360°.  The  plate  carries 
a  frame  supporting  the 
axis  of  a  graduated  circle; 
this  axis  is  perpendicular 
to  the  upright  axis,  and  the 
circle  is  graduated  from  0° 
to  90°  in  opposite  directions. 
Attached  to  the 
circle  is  a  tele- 
scope whose  op- 
tical axis  is  in 
the  plane  of  the 
circle.  The  cir- 
cle is  read  by 
a  pointer  which 
is  fixed  to  the 

frame  carrying  its  axis  and  reads  0°  when  the  optical  axis  of  the 
telescope  is  perpendicular  to  the  upright  axis.  A  level  is  attached 
to  the  telescope  so  that  the  bubble  is  in  the  center  of  its  tube  when 
the  telescope  is  horizontal.  In  what  follows,  all  these  adjustments 
are  supposed  to  be  properly  made  by  the  maker. 


FIG.  31 


36 


LABORATORY    ASTRONOMY 


ADJUSTMENT   OF   THE    ALTAZIMUTH 

If  the  altazimuth  is  so  adjusted  that  the  upright  axis  is  exactly 
vertical,  and  if  we  know  the  reading  of  the  horizontal  circle  when 
the  vertical  circle  lies  in  the  meridian,  we  may  determine  the 
position  of  a  heavenly  body  at  any  time  by  pointing  the  telescope 
upon  it  and  reading  the  two  circles.  The  difference  between  the 
reading  of  the  horizontal  circle  and  its  meridian  reading  is  the  azi- 
muth, and  the  reading  of  the  vertical  circle  is  the  altitude  of  the 
body.  Before  proceeding  to  the  observation  of  stars,  it  will  be  well 
to  repeat  our  observations  on  the  sun,  using  this  instrument,  and 
making  them  in  such  a  manner  that  we  may  at  the  same  time  get  a 
very  exact  determination  of  the  meridian  reading  by  the  method 
suggested  on  page  14. 

Place  the  instrument  upon  the  table  used  for  the  sun  observation ; 
bring  the  reading  of  each  circle  to  0° ;  and  turn  the  whole  instru- 
ment in  a  horizontal  plane  until  the  telescope  points  approximately 
south,  using  the  meridian  determination  obtained  from  the  shadow 
observations.  One  leveling  screw  will  then  be  nearly  in  the  meridian 
of  the  center  of  the  instrument,  while  the  two  others  will  lie 
in  an  east  and  west  line.  Bring  the  level  bubble  to  the  middle 
of  its  tube  by  turning  the  north  leveling  screw;  then  set  the 
telescope  pointing  east;  and  "set"  the  level  by  turning  the  east 
and  west  screws  in  opposite  directions.  Be  careful  to  turn  them 
equally  ;  this  can  be  done  by  taking  one  leveling  screw  between  the 


FIG.  32 


finger  and  thumb  of  each  hand,  holding  them,  firmly,  and  turning 
them  in  opposite  directions  by  moving  the  elbows  to  or  from  the 
body  by  the  same  amount.  Turn  the  telescope  north,  and  the  bubble 


THE    DIURNAL    MOTION    OF    THE    STARS  37 

should  remain  in  place  j  if  it  does  not,  adjust  the  north  screw.  The 
instrument  is  very  easily  and  quickly  adjusted  by  this  method. 
The  upright  axis  is  vertical  when  the  telescope  can  be  turned  about 
it  into  any  position  without  displacing  the  bubble. 

Determination  of  the  Meridian  and  Time  of  Apparent  Noon.  —  After 
completing  the  adjustment  of  the  instrument,  the  reading  of  the  circle 


FIG.  33 

when  the  telescope  is  in  the  meridian  is  determined  as  follows :  Point 
the  telescope  upon  the  sun  approximately.  Place  a  sheet  of  paper  or 
a  card  behind  it,  and  turn  the  telescope  about  the  vertical  axis  until 
the  shadow  of  the  vertical  circle  is  reduced  to  its  smallest  dimensions 
and  appears  as  a  broad  straight  line.  By  moving  the  telescope 
about  the  horizontal  axis,  bring  the  shadow  of  the  tube  to  the  form 
of  a  circle ;  in  this  circle  will  appear  a  blurred  disk  of  light.  Draw 
the  card  about  10  inches  back  from  the  eyepiece,  and  pull  out  the 
latter  nearly  -J-  of  an  inch  from  its  position  when  focused  on  distant 
objects  and  the  disk  of  light  becomes  nearly  sharp ;  complete  the 
focusing  of  this  image  of  the  sun  by  moving  the  card  to  or  from 
the  eyepiece.  The  distance  of  the  card  and  the  drawing  out  of  the 
eyepiece  should  be  such  that  the  sun's  image  shall  be  about  -J-  to  j- 
of  an  inch  in  diameter.  Now  move  the  telescope  until  the  image  is 
centered  in  the  shadow  of  the  telescope  tube,  note  the  time,  and  read 
both  circles ;  this  observation  fixes  the  altitude  and  azimuth  of  the 


38  LABORATORY    ASTRONOMY 

sun.  For  determining  the  meridian  it  is  not  necessary  that  the 
time  should  be  noted,  but  it  will  be  convenient  to  use  these  obser- 
vations for  a  repetition  of  the  determination  of  the  sun's  path,  deter- 
mining the  altitudes  and  azimuths  by  this  more  accurate  method. 

This  observation  should  be  made  at  least  as  early  as  9  A.M. 
Now  increase  the  reading  of  the  vertical  circle  to  the  next  exact 
number  of  degrees,  and  follow  the  sun  by  moving  the  telescope 
about  the  vertical  axis.  After  a  few  minutes  the  sun  will  be  again 
centered  by  this  process.  Note  the  time,  and  read  the  horizontal 
circle.  Increase  the  reading  of  the  vertical  circle  again  by  one 
degree  to  make  another  observation,  and  so  on  for  half  an  hour. 
Observations  may  be  made  at  one-half  degree  intervals  of  altitude, 
but  those  upon  exact  divisions  will  evidently  be  more  accurate.  If 
circumstances  admit,  observations  may  be  made,  during  the  period 
of  two  hours  before  and  after  noon,  for  the  purpose  of  plotting  the 
sun's  path ;  but,  owing  to  the  slow  change  of  altitude  in  that  time, 
the  corresponding  azimuths  are  not  well  determined,  and  they  will 
be  nearly  useless  for  placing  the  instrument  in  the  meridian. 

Some  time  in  the  afternoon,  as  the  descending  sun  approaches  the 
altitude  last  observed  in  the  forenoon,  set  the  vertical  circle  upon  the 
reading  corresponding  to  that  observation,  and  repeat  the  series  in 
inverse  order  ;  that  is,  decrease  the  readings  of  altitude  by  one  degree 
each  time,  and  note  the  time  and  the  reading  of  the  horizontal  circle 
when  the  sun  is  in  the  axis  of  the  telescope  at  each  successive 
altitude. 

Since  equal  altitudes  correspond  to  equal  azimuths  (see  page  14), 
east  and  west  of  the  meridian,  the  difference  of  the  horizontal  read- 
ings is  twice  the  azimuth  at  either  of  the  two  corresponding  obser- 
vations (360°  must  be  added  to  the  western  reading,  if,  as  will 
generally  be  the  case,  the  0°  point  lies  between  the  two  readings). 
Therefore,  one-half  this  difference  added  to  the  lesser  or  subtracted 
from  the  greater  reading  gives  the  meridian  reading.  The  same 
value  is  more  easily  found  by  taking  half  the  sum  of  the  two  read- 
ings. In  the  same  way  one-half  the  interval  of  time  between  the 
two  observations  added  to  the  time  of  the  first  reading  gives  the 
watch  time  of  the  sun's  meridian  passage,  or  apparent  noon,  as  it 
is  called. 


THE    DIURNAL    MOTION    OF    THE    STARS 


39 


Each  pair  of  observations  gives  the  value  of  the  meridian  reading 
and  of  the  watch  time  of  apparent  noon ;  their  accordance  will  give 
an  idea  of  the  accuracy  of  the  observations. 

The  following  observations  of  the  sun  were  made  March  8,  1900, 
with  an  instrument  similar  to  that  shown  in  Fig.  33. 


TIME 

ALTITUDE 

HORIZONTAL 
CIRCLE 

TIME 

ALTITUDE 

HORIZONTAL 
CIRCLE 

1     8* 

54™ 

37" 

27°. 

5 

307°.  6 

9 

2h  32m 

10« 

31°.0 

47°.  7 

2     8 

58 

10 

28  . 

0 

308  .4 

10 

2    36 

30 

30  .5 

48  .7 

3    9 

1 

42 

28  . 

5 

SM9  .2 

11 

2    39 

45 

30  .0 

49  .45 

4     9 

4 

51 

29  . 

0 

310  .0 

12 

2    43 

27 

29.5 

50.35 

5     9 

9 

5 

29  . 

5 

310  .9 

13 

2    47 

0 

29  .0 

51  .15 

6     9 

12 

20 

30. 

0 

311  .8 

14 

2    50 

17 

28  .5 

51  .95 

7     9 

15 

35 

30  . 

5 

312  .6 

15 

2    54 

7 

28.0 

52  .85 

8     9 

19 

37 

31  . 

0 

313  .45 

16 

2    57 

33 

27  .5 

53  .6 

The  1st  and  16th  of  these  observations  give  for  the  meridian 
reading  %  [307.6  +  (53.60  +  360)]  =  360°.60,  and  for  the  correspond- 
ing watch  time  £[8  54  37  +  (2  57  33  +  12h)]  =  llh  56m  5s. 

Taking  the  corresponding  A.M.  and  P.M.  observations  in  this  man- 
ner, we  find  for  the  eight  pairs  of  observations  above  the  following 

values. 

MERIDIAN  READING 
360°.  6 
360  .625 
.575 
.575 
.625 
.625 
.65 
.575 


ALTITUDE 
27°.  5 
28  .0 

28  .5 

29  .0 
29.5 

30  .0 

30  .5 

31  .0 
mean 


360 
360 
360 
360 
360 
360 
360.61 


WATCH  TIME  OF  NOON 
llh  561"  5.QS 
56  8.5 
55  59.5 

55  55.5 

56  16.0 
56  2.5 
56  2.5 
55  53.5 


11  56  2.9 


The  agreement  of  these  results  is  closer  than  will  usually  be 
obtained,  the  observations  being  made  by  a  skilled  observer  and  the 
angles  carefully  read  by  means  of  a  pocket  lens,  which  in  many 
cases  enabled  readings  to  be  made  to  0°.05 ;  any  reading  such  as  that 
of  the  8th  observation,  where  the  value  was  estimated  to  lie  between 
two  tenths,  being  recorded  as  lying  halfway  between  them.  This 
practice  adds  little  to  the  accuracy  if  several  observations  are  made, 
and  is  not  to  be  recommended  to  beginners. 


40  LABORATORY    ASTRONOMY 

MERIDIAN   MARK 

It  will  be  convenient  to  fix  a  meridian  mark  for  future  use.  This 
may  be  done  by  fixing  the  telescope  at  the  meridian  reading,  turning 
it  down  to  the  horizontal  position,  and  placing  some  object  (as  a 
stake)  at  as  great  a  distance  as  possible,  so  that  it  may  mark  the 
line  of  the  axis  of  the  telescope  when  in  the  meridian.  A  mark  on 
a  fence  or  building  will  serve  if  at  a  greater  distance  than  50  feet, 
though  a  still  greater  distance  is  desirable.  For  setting  the  tele- 
scope upon  the  mark,  it  is  convenient  to  have  two  wires  crossing  in 
the  center  of  the  field  of  view,  but  the  setting  may  be  made  within 
0°.l  without  this  aid.  Having  established  such  a  mark,  set  the 
horizontal  circle  at  0°,  and  move  the  whole  base  of  the  instrument 
until  the  telescope  points  upon  the  meridian  mark.  Level  carefully ; 
then  set  the  telescope  again,  if  the  operation  of  leveling  has  caused 
it  to  move  from  the  meridian  mark ;  level  again,  and  by  repeating 
this  process  adjust  the  instrument  so  that  it  is  level  and  that  the 
telescope  is  in  the  meridian.  Then  press  hard  on  the  leveling 
screws,  and  make  dents  by  which  the  instrument  can  be  brought 
into  the  same  position  at  any  future  time. 

After  the  A.M.  and  P.M.  observations  recorded  above,  the  tele- 
scope was  pointed  upon  a  meridian  mark  established  by  observations 
made  with  the  shadow  of  a  pin,  and  the  reading  of  the  horizontal 
circle  was  359°.8.  The  mark  was  then  shifted  about  a  foot  toward 
the  west,  and  the  telescope  again  pointed  upon  it.  As  the  reading 
pf  the  circle  was  then  3 60°. 6,  it  may  be  assumed  that  the  mark  was 
now  very  nearly  in  the  meridian. 

If  circumstances  are  such  that  no  point  of  reference  in  the  meridian 
is  available,  it  will  be  necessary,  after  determining  the  meridian 
readings  by  the  sun,  to  set  the  telescope  upon  some  well-defined  object 
in  or  near  the  horizontal  plane  and  read  the  circle.  The  difference 
between  this  reading  and  the  meridian  reading  will  be  the  azimuth 
of  the  object.  Set  the  pointer  of  the  horizontal  circle  to  this  value, 
and  set  the  telescope  upon  the  reference  mark  by  moving  the  whole 
base  as  before.  If  the  pointer  of  the  circle  is  now  brought  to  0°,  the 
telescope  will  evidently  be  in  the  meridian ;  and  the  position  is  to 
be  fixed  by  making  dents  with  the  leveling  screws  as  before. 


THE   DIURNAL    MOTION    OF    THE    STARS  41 

CHOICE   OF  STARS 

We  are  now  ready  to  begin  observations  of  the  stars. 

The  most  familiar  group  of  stars  in  the  heavens  is,  no  doubt,  that 
part  of  the  Great  Bear  which  is  variously  called  the  Dipper,  Charles's 
Wain,  or  the  Plough. 

At  the  beginning  of  October,  at  8  o'clock  in  the  evening,  an 
observer  anywhere  in  the  United  States  will  see  the  Dipper  at  an 
altitude  between  10°  and  30°  above  the  N.W.  horizon.  Set  the 
telescope  upon  that  star  which  is  nearest  the  north  point  of  the 
horizon ;  read  both  circles  to  determine  its  altitude  and  azimuth, 
and  note  the  time.  Even  if  the  telescope  is  provided  with  cross- 
hairs, the  illumination  of  the  light  of  the  sky  will  not  be  suffi- 
cient to  render  them  visible :  but  sufficient  accuracy  in  pointing  is 
obtained  by  placing  the  star  at  the  estimated  center  of  the  field. 
Observe  in  succession  the  altitude  and  azimuth  of  the  other  six 
stars  forming  the  Dipper,  noting  the  time  in  each  case. 

Using  the  Dipper  as  a  starting  point,  we  will  now  identify  and 
observe  a  few  other  stars.*  The  total  length  of  the  Dipper  is  about 
25°.  Following  approximately  a  line  drawn  joining  the  last  two 
stars  of  the  handle  of  the  Dipper,  at  a  distance  of  about  30°,  we 
come  to  a  bright  star  of  a  strong  red  color,  much  the  brightest  in 
that  portion  of  the  heavens  ;  this  is  Arcturus.  Observe  its  altitude 
and  azimuth,  and  note  the  time  as  before.  Almost  directly  over- 
head, too  high  to  be  conveniently  observed  at  this  time,  is  a  bril- 
liant white  star,  Vega  (a  Lyrse).  A  little  east  of  south  from  Vega, 
at  an  altitude  of  about  60°,  is  a  group  of  three  stars  forming  a  line 
about  5°  in  length.  The  central  and  brightest  star  of  the  three  is 
Altair  (a  Aquilse),  and  its  position  should  be  observed. 

Diurnal  Paths  of  the  Stars.  —  Proceed  in  this  way  for  about  an 
hour,  observing  also,  if  time  permits,  the  group  of  five  stars  whose 
middle  is  at  azimuth  220°  and  altitude  35°.  This  is  the  constella- 
tion of  Cassiopeia.  Another  interesting  asterism  will  be  found  — 
supposing  that  by  this  time  it  is  9  o'clock  —  at  azimuth  270°  and 
altitude  45°,  consisting  of  four  stars  of  about  equal  magnitude, 

*  Many  of  the  latest  text-books  on  astronomy  contain  small  star  maps  which 
are  valuable  aids  in  the  identification  of  the  less  conspicuous  groups. 


42  LABORATORY  ASTROHOMY 

placed  at  the  corners  of  a  quadrilateral  whose  sides  are  about  15d 
in  length,  and  forming  what  is  called  the  Square  of  Pegasus. 

It  is  convenient  as  an  aid  in  identification  to  note  in  each  case  the 
magnitude  of  the  star  observed.  As  a  rough  standard  of  compari- 
son, it  may  be  remembered  that  the  six  bright  stars  of  the  Dipper 
are  of  about  the  second  magnitude;  that  at  the  junction  of  the 
handle  and  bowl  is  of  the  fourth.  The  three  stars  in  Aquila  are  of 
the  first,  third,  and  fourth  magnitudes.  Vega  and  Arcturus  are 
each  larger  than  an  average  first  magnitude  star.  The  brightest 
stars  in  the  constellation  Cassiopeia  and  in  the  Square  of  Pegasus 
range  from  the  second  to  the  third  magnitude. 

The  little  quadrilateral  of  fourth  magnitude  stars  about  15°  east 
of  Altair  and  known  as  Delphinus,  or  vulgarly  as  Job's  Coffin,  may 
be  observed. 

At  the  expiration  of  an  hour,  set  again  upon  the  Dipper  stars  and 
repeat  the  series,  going  through  the  same  list  in  the  same  order.  Arc- 
turus will  have  sunk  so  low  in  a  couple  of  hours  as  to  be  beyond  the 
reach  of  observation,  even  if  the  place  of  observation  affords  a  clear 
view  of  the  horizon.  Vega,  however,  will  be  less  difficult  to  observe, 
and  may  be  now  added  to  the  list.  We  should  not  omit  to  make 
an  observation  of  the  pole  star,  which,  as  its  name  indicates,  may 
be  found  near  the  pole  and  can  be  easily  found,  since  the  azimuth 
of  the  pole  is  180°,  and  its  altitude  is  equal  to  the  latitude  of  the 
place. 

From  the  observed  values  of  altitude  and  azimuth  plot  the  suc- 
cessive places  on  the  hemisphere  exactly  as  in  the  case  of  the  sun, 
and  thus  represent  upon  the  hemisphere  the  paths  of  a  number  of 
stars  in  various  parts  of  the  heavens.  It  will  be  found  that  these 
paths  are  all  circles  of  various  dimensions,  and  that  the  circles  are 
all  parallel  to  the  equator,  as  determined  from  the  sun  observations, 
that  is,  they  have  the  same  pole  as  the  diurnal  circles  of  the  sun. 

At  this  stage  it  is  a  good  plan  to  devote  some  attention  to  the 
representation  of  the  various  results  as  shown  on  the  hemisphere, 
by  means  of  figures  on  a  plane  surface,  that  is,  to  make  careful  free- 
hand drawings  of  the  hemisphere  and  the  circles  which  have  now 
been  drawn  upon  it  as  seen  from  various  points  of  view.  This  is 
an  important  aid  to  the  understanding  of  the  diagrams  by  which  it 


THE    DIURNAL    MOTION    OF    THE    STARS  43 


is  necessary  to  explain  the  statement  and  solution  of  astronomical 
problems  ;  with  this  purpose  in  view  the  drawings  should  be  lettered 
and  the  definitions  of  the  various  points  and  lines  written  under 
them. 

ROTATION   OF    THE    SPHERE    AS    A   WHOLE 

So  far  the  result  of  our  observations  is  to  show  that  the  heavenly 
bodies  appear  to  move  as  they  would  if  they  were  all  attached  in 
some  way  to  the  same  spherical  shell  surrounding  the  earth,  and 
were  carried  about  by  a  common  revolution,  as  if  the  shell  rotated 
on  a  fixed  axis,  passing  through  the  point  of  observation.  The  sun 
may  be  conceived  as  carried  by  the  same  shell,  but  observations  at 
different  dates  show  that  its  place  on  the  shell  must  slowly  change, 
since  its  declination  changes  slightly  from  day  to  day. 

If  these  observations  on  the  stars  are  repeated  ten  days  or  one 
hundred  days  later,  we  shall  find  that  the  declinations  determined 
from  them  are  the  same ;  that  is,  the  declinations  of  the  diurnal 
paths  of  the  stars  do  not  change  like  that  of  the  sun.  It  will  appear 
also  that,  as  in  the  case  of  the  sun,  equal  arcs  of  the  diurnal  circle 
and  consequently  equal  hour-angles  are  described  in  equal  times. 
It  follows  from  this,  of  course,  that  stars  nearer  the  pole  will  appear 
to  move  more  slowly,  since  they  describe  paths  which  are  shorter 
when  measured  in  degrees  of  a  great  circle,  as  may  be  shown  by 
measuring  the  diurnal  circles  on  the  hemisphere  by  a  flexible  milli- 
meter scale,  1  mm.  being  equal  to  1°  of  a  great  circle  on  our 
hemisphere. 

If  the  field  of  view  of  our  telescope  is  5°,  a  star  on  the  equinoc- 
tial will  be  carried  across  its  center  by  the  diurnal  motion  in  20 
minutes,  while  a  star  at  a  declination  of  60°  will  remain  in  the  field 
for  twice  that  time,  since  its  diurnal  circle  is  only  half  as  large  as 
the  equinoctial  and  an  angular  motion  of  10°  of  its  diurnal  circle  is 
only  5°  of  great  circle.  Since  the  declinations  of  the  stars  do  not 
change,  it  is  unnecessary  to  make  our  observations  of  the  stars  on 
the  same  night ;  or,  rather,  observations  made  on  different  nights 
may  be  plotted  as  if  made  on  the  same  night.  We  may  thus  obtain 
extensions  of  the  diurnal-  circles  by  working  early  on  one  evening 
and  at  later  hours  of  the  night  on  following  occasions. 


44  LABORATORY   ASTRONOMY 

POSITIONS    FIXED    BY    HOUR-ANGLE    AND    DECLINATION; 
THE   EQUATORIAL 

It  is  evident  that  we  have,  in  the  hour-angles  and  declinations 
of  the  stars,  another  system  of  coordinates  on  the  celestial  sphere 
by  means  of  which  their  position  may  be  fixed.  The  altitude  and 
azimuth  refer  the  position  of  the  star  to  the  meridian  and  to  the 
horizon ;  while  the  hour-angle  and  declination  refer  its  position  to 
the  meridian  and  the  equator.  We  have  hitherto  found  it  more 
convenient  to  deal  with  the  first  set  of  coordinates,  but  it  is  often 
desirable  to  determine  the  hour-angle  and  declination  of  a  body  by 
direct  observation,  and  this  may  be  done  by  means  of  an  instrument 
similar  to  the  altazimuth  but  with  the  upright  axis  pointed  to  the 
pole  of  the  heavens,  so  that  the  horizontal  circle  lies  in  the  plane  of 
the  equator.  With  this  instrument  the  angles  read  off  on  the  circle 
which  is  directly  attached  to  the  telescope  measure  distances  along 
the  hour-circle,  perpendicular  to  the  equator,  i.e.,  declinations,  while 
an  angle  read  off  on  the  other  circle  measures  the  angle  between  the 
meridian  and  the  hour-circle  of  the  star  at  which  the  telescope  points, 
and  is  therefore  the  star's  hour-angle.  The  two  circles  are  there- 
fore appropriately  called  the  declination  circle  and  the  hour-circle 
of  the  instrument.  As  these  terms  are  used  with  another  meaning 
as  applied  to  circles  on  the  celestial  sphere,  it  would  seem  that  there 
might  be  confusion  from  their  use  in  this  sense,  but  in  practice  it 
is  never  doubtful  whether  "  circle "  means  the  graduated  circle  of 
an  instrument  or  a  geometrical  circle  on  the  surface  of  the  sphere. 

It  is  here  supposed  that  the  instrument  has  been  so  adjusted 
that  both  circles  read  0°  when  the  telescope  is  in  the  plane  of  the 
meridian  and  points  at  the  equator.  An  instrument  so  mounted  is 
called  an  equatorial  instrument.  Our  altazimuth  is  adapted  to  this 
purpose  by  constructing  the  base  so  that  it  may  be  revolved  about  a 
horizontal  axis  perpendicular  to  the  plane  in  which  the  altitude 
circle  lies  when  the  azimuth  circle  reads  0°.  If,  then,  it  has  been 
placed  in  the  meridian  by  the  observation  of  equal  altitudes  as 
before  described,  it  may  be  inclined  about  this  latter  axis  through 
an  angle  equal  to  the  complement  of  the  latitude,  and  thus  brought 
into  the  proper  position  for  observing  declination  and  hour-angle 


THE    DIURNAL    MOTION    OF    THE    STARS 


45 


directly.  An  instrument  so  constructed  is  called  a  "universal" 
equatorial.  To  adjust  the  universal  equatorial  so  that  the  axis 
points  to  the  pole,  adjust  it  as  an  altazimuth  with  both  circles 
reading  0°  and  level  it  with  the  telescope  in  the  meridian  pointing 
south.  Depress  the  telescope  till  the  reading  of  the  vertical  circle 
equals  the  co-latitude.  Tip  the  whole  instrument  so  as  to  incline 
the  vertical  axis  to- 
ward the  north  till 
the  bubble  plays  and 
the  telescope  is  hori- 
zontal ;  to  do  this  the 
vertical  axis  must 
have  been  tipped 
back  through  an 
angle  equal  to  the  co- 
latitude,  and  it  will 
be  in  proper  adjust- 
ment directed  toward 
a  point  in  the  merid- 
ian whose  altitude 
is  equal  to  the  lati- 
tude. (Fig.  34  shows 
the  instrument 
adjusted  for  latitude 
45°.) 

A  notch  should  be  cut  in  the  iron  arc  at  the  bottom  of  the  coun- 
terpoise, into  which  the  spring-catch  may  slip  when  the  adjustment 
is  correct,  so  that  the  instrument  may  be  quickly  changed  from  one 
position  to  the  other.  If  the  notch  is  not  quite  correctly  placed, 
the  final  adjustment  may  be  made  by  a  slight  motion  of  the  north 
leveling  screw  to  bring  the  level  exactly  into  the  horizontal  position, 
the  vertical  circle  having  been  set  to  the  co-latitude  for  this  purpose. 

The  proper  adjustment  of  the  altazimuth  is  simpler,  since  it 
depends  only  on  the  use  of  the  level,  while  to  place  an  equatorial 
instrument  in  position  we  must  know  the  latitude  as  well.  On  coin- 
paring  the  two  systems  of  coordinates,  it  is  clear  that,  while  the 
altitude  and  azimuth  both  change  continuously,  but  not  uniformly 


FIG.  34 


46 


LABORATORY    ASTRONOMY 


with  the  time,  the  hour-angle  changes  uniformly  with  the  time, 
and  the  declination  remains  the  same.  One  advantage  of  the 
latter  system  of  coordinates  is  that  in  repeating  our  observations 
on  the  same  star  after  the  lapse  of  an  hour,  we  need  only  set 
the  declination  circle  to  the  previously  observed  declination,  and 
set  the  hour-circle  at  a  reading  obtained  by  adding  to  the  former 
setting  the  elapsed  time  in  hours  reduced  to  degrees  by  multiply- 
ing by  15 ;  we  shall  then  pick  up  the  star  without  difficulty.  This 
is  an  important  aid  in  identifying  stars,  which  has  no  counterpart 
in  the  use  of  the  altazimuth,  and  we  shall  henceforth  use  this 
method  of  observation  in  preference  to  the  other. 


CHAPTEE  IV 
THE  COMPLETE   SPHERE   OF  THE  HEAVENS 

THE  study  of  the  motions  of  the  sun,  moon,  and  stars  has  thus, 
far  led  to  the  conclusion  that  their  courses  above  the  plane  of  the 
horizon  can  be  perfectly  represented  by  assuming  the  daily  rotation 
from  east  to  west  of  a  sphere  to  which  they  are  attached,  or  a  rota- 
tion of  the  earth  itself  from  west  to  east  about  an  axis  lying  in  the 
meridian  and  inclined  to  the  horizon  at  an  angle  equal  to  the  latitude 
of  the  place  of  observation,  while  the  sun  moves  slowly  to  and  from 
the  equator,  and  the  moon,  like  the  sun,  changes  its  declination  con- 
tinually, and  has  also  a  motion  toward  the  east  on  the  sphere  at  a 
rate  of  about  13°  in  each  24  hours.  The  combination  of  the  two 
motions  of  the  moon  causes  it  to  describe  a  path  which  will  be  more 
fully  discussed  later.  We  shall  now  begin  to  observe  the  sun,  to 
see  if  its  motion  among  the  stars  resembles  that  of  the  moon  in 
having  an  east  and  west  component  in  addition  to  its  motion  in 
declination. 

The  motion  of  the  moon  can  be  directly  referred  to  the  stars,  since 
both  are  visible  at  the  same  time,  although  the  illumination  of  the 
dust  of  our  atmosphere,  by  strong  moonlight,  cuts  us  off  from 
the  use  of  the  smaller  stars,  which  cannot  be  seen  except  when 
contrasted  with  a  perfectly  dark  background. 

The  illumination  produced  by  the  sun,  however,  is  so  strong  that 
it  completely  blots  out  even  the  brightest  stars,  so  that  we  cannot 
apply  either  of  the  methods  that  we  have  employed  in  observing 
the  moon. 

We  are  only  able  to  see  the  stars,  of  course,  when  they  are  above 
the  plane  of  the  horizon,  but  it  is  natural  to  suppose  that  they  con- 
tinue the  same  course  below  the  horizon  from  their  points  of  setting 
to  those  of  their  rising.  This  inference  is  confirmed  by  the  fact  that 
some  of  the  bright  stars  which  set  within  a  few  degrees  of  the  north 
point  of  the  horizon,  and  which  we  infer  complete  their  course  below 

47 


48  LABORATORY    ASTRONOMY 

the  horizon,  may  be  seen  actually  to  do  so  by  an  observer  at  a  point 
on  the  earth  some  degrees  farther  north,  from  which  they  may  be 
observed  throughout  the  whole  of  their  courses.  In  the  case  of  the 
sun,  the  following  facts  lead  to  the  same  conclusion.  Immediately 
after  sunset  a  twilight  glow  is  seen  in  the  west  whose  intensity  is 
greatest  at  the  point  where  the  sun  has  just  set.  This  glow  appears  to 
pass  along  the  horizon  towards  the  north,  and  its  point  of  greatest 
intensity  is  observed  to  be  directly  over  the  position  which  the  sun 
would  occupy  in  the  continuation  of  its  path  below  the  horizon,  on 
the  assumption  that  it  continues  to  move  uniformly  in  that  path. 
In  high  latitudes  this  change  of  position  in  the  twilight  arch  can  be 
followed  completely  around  from  the  point  of  sunset  to  the  point  of 
sunrise,  the  highest  point  being  due  north  at  midnight.  It  is  impos- 
sible not  to  believe  that  the  sun  is  actually  there,  though  concealed 
from  our  sight  by  the  intervening  earth.  (Of  course,  too,  it  is  now 
generally  known  that  in  very  high  latitudes  the  sun  at  midsummer 
is  visible  throughout  its  diurnal  course.)  As  the  sun  sinks  farther, 
the  light  of  the  sky  decreases,  the  brighter  stars  begin  to  appear, 
and  it  is  clearly  impossible  to  resist  the  conclusion  that  they  have 
been  in  position  during  the  daylight,  but  simply  blotted  out  by  the 
overwhelming  light  of  the  sun. 

OBSERVATIONS   WITH   THE   EQUATORIAL 

When  we  have  fixed  the  idea  that  the  heavenly  sphere  revolves 
as  a  whole,  carrying  with  it  in  a  general  sense  all  the  bodies  that  we 
observe,  the  next  step  is  to  devise  some  means  of  locating  the  dif- 
ferent bodies  in  their  proper  relative  positions  on  the  sphere.  For 
this  purpose  the  equatorial  instrument  furnishes  us  with  an  admi- 
rable means  of  observation.  The  relative  position  of  two  stars  is 
completely  fixed  when  we  know  the  position  of  their  parallels  of 
declination  and  their  hour-circles,  since  the  place  of  each  star  is 
at  the  intersection  of  these  two  circles. 

Since  an  observation  with  the  equatorial  gives  directly  the  decli- 
nation and  hour-angle  of  a  star,  the  method  of  fixing  the  relative 
position  of  two  stars,  A  and  B,  is  as  follows : 

Point  the  telescope  at  A,  and  read  the  circles ;  then  set  on  B,  and 


THE    COMPLETE    SPHERE    OF    THE    HEAVENS  49 

read  the  circles ;  then  again  on  A,  and  read  the  circles,  taking  care 
that  the  interval  between  the  first  and  second  observations  shall  be 
as  nearly  as  possible  equal  to  the  interval  between  the  second  and 
third.  Obviously  the  mean  of  the  two  readings  of  the  hour-circle 
at  the  pointings  upon  A  gives  the  hour-angle  of  A  at  the  time  when 
B  was  observed,  since  the  star's  hour-angle  changes  uniformly. 
The  difference  between  this  mean  and  the  reading  of  the  hour- 
circle  when  the  pointing  was  made  upon  B  is,  therefore,  the  dif- 
ference between  the  hour-angles  of  the  stars  at  the  time  of  that 
observation ;  and  this  fixes  the  relative  position  of  their  hour-circles, 
since  this  difference  is  the  arc  of  the  equator  included  between 
them ;  their  declinations  are  given  by  the  readings  of  the  declina- 
tion circles,  and  thus  the  relative  position  of  the  two  stars  is 
completely  known. 

As  an  illustration  of  this  method,  we  may  take  the  following 
example : 

With  the  telescope  pointed  at  A,  the  readings  of  the  hour-circle 
and  declination  circle  were  68°.2  and  15°.l,  respectively.  The 
telescope  was  then  pointed  at  B,  and  the  circles  read  85°.9,  28°.l, 
and  finally  upon  A,  the  readings  being  69°.l,  15°.l ;  the  intervals 
were  nearly  the  same,  as  will  usually  be  the  case,  unless  there  is 
some  difficulty  in  finding  the  second  star.  Of  course  the  first  star 
can  be  re-found  by  the  readings  at  the  first  observation ;  indeed, 
if  the  intervals  are  plainly  unequal,  a  repetition  of  the  observation 
may  always  be  made  at  equal  intervals  by  setting  the  circles  for 
each  star  so  that  no  time  is  lost  in  finding. 

From  the  above  observations  we  infer  .that  when  the  hour-angle 
of  B  was  85°.  9,  that  of  A  was  68°.  65 ;  and,  therefore,  that  the  hour- 
circles  of  the  two  stars  cut  the  equator  at  points  17°.25  apart; 
the  hour-circle  of  B  being  to  the  west  of  that  of  A,  so  that  B  comes 
to  the  meridian  earlier,  or  "precedes"  A. 

It  may  be  noted  that  the  observations  apparently  occupied  a  little 
less  than  4  minutes,  since  in  the  whole  interval  the  hour-angle  of  A 
changed  by  0°.9. 


50 


LABORATORY    ASTRONOMY 


USE   OF   A   CLOCK   WITH   THE   EQUATORIAL 

If  the  intervals  between  the  observations  are  not  exactly  equal, 
it  will  still  be  easy  to  fix  the  hour-angle  of  A  at  the  time  of  the 
observation  on  B  if  the  ratio  of  the  intervals  is  known;  if,  for 
instance,  the  first  observation  of  A  gives  an  hour-angle  of  25°. 3,  and 
the  later  observation  an  hour-angle  of  26°.3,  while  the  intervals 
are  lm  between  the  first  and  second  observations,  and  3m  between 
the  second  and  third,  the  hour-angle  of  A  at  the  second  observa- 
tion was  obviously  25°. 3  +  0°.25.  We  may  thus  obtain  by  "  inter- 
polation "  the  hour-angle  of  A  at  any  known  fraction  of  the  interval. 
Plainly  it  is  an  advantage  to  note  the  time  of  each  observation  for 
this  purpose,  as  in  the  following  observations,  which  were  made 
Feb.  5,  1900,  for  the  purpose  of  determining  the  relative  positions 
of  the  stars  forming  the  Square  of  Pegasus. 


STAR 

WATCH  TIME 

DECL.  CIRCLE 

HOUR-ClKCLE 

1   7  Pegasi 

7h    14m   Os 

+  15°.  2 

66°.  3 

2   a  Pegasi 

15     0 

.  +  15  .2 

83  .6 

3  ft  Pegasi 

16      15 

+  28  .1 

84  .1 

4   a  AndromedaB 

17      10 

+  29  .0 

68  .3 

5   7  Pegasi 

18     30 

+  15  .2 

67  .6 

6  7  Pegasi 

21     30 

+  15  .1 

(69  .2) 

7   a  AndromedaB 

22     30 

+  29  .1 

69  .6 

8  ft  Pegasi 

23     30 

+  28  .1 

85  .9 

9  oJPegasi 

24     20 

+  15  .3 

86  .0 

10  7  Pegasi 

25     30 

+  15  .1 

69  .1 

11   7  Pegasi 

27     30 

+  15  .1 

69  .6 

The  observations  here  follow  each  other  rapidly.  They  were  made 
by  an  experienced  observer,  and  the  arrangement  of  the  stars  is  such 
that,  after  setting  y  Pegasi,  a  Pegasi  is  brought  into  the  field  by 
moving  the  telescope  about  the  hour-axis  only ;  .we  pass  to  ft  Pegasi 
by  motion  around  the  declination  axis  only,  to  a  Andromedae  by 
motion  about  the  hour-axis,  and  back  to  y  Pegasi  by  rotation  about 
the  declination  axis ;  so  that  the  stars  are  found  more  quickly 
than  if  both  axes  must  be  altered  in  position  at  each  change ;  in 
observations  6  to  10  the  series  is  observed  in  reversed  order. 


THE    COMPLETE    SPHERE    OF    THE    HEAVENS  51 

If  the  instrument  was  correctly  adjusted,  the  declination  of  the 
four  stars  was  as  follows  :  y  Fegasi  + 15°.14,  a  Pegasi  15°.25, 
ft  Pegasi  28°.l,  a  Andromedse  29°. 05,  each  being  determined  as  the 
mean  of  all  the  observations  made  upon  the  star. 

The  first  advantage  of  the  recorded  times  is  to  show  that  the 
reading  of  the  hour-circle  in  6  was  an  error,  probably  for  68°.2,  as 
we  see  by  comparison  with  the  other  values  of  the  hour-angle  of 
y  Pegasi,  which  increase  uniformly  about  1°  in  each  4  minutes.  It 
will  be  better,  however,  to  reject  the  observation  entirely,  as  it  is 
not  necessary  to  use  it  for  the  first  set  of  observations  1  to  5,  which 
we  will  now  discuss. 

By  interpolation  between  1  and  5  we  find  that  the  hour-angle  of 
y  Pegasi  at  7h  15m  0s  was  f  of  1°.3  greater  than  66°.3,  or  66°.59; 
at  7h  16m  15s  it  was  £  of  1°.3  greater  than  66°.3,  or  66°.95;  and  at 
7h  17m  10s  it  was  _8_p_  Of  i°.3  iess  than  67°.6,  or  67°.21.  As  the 
hour-angles  of  the  other  stars  were  observed  at  these  times,  we 
can  at  once  find  the  differences  of  their  hour-angles  from  that  of 
y  Pegasi,  which  are  as  follows  :  a  Pegasi,  17°.01 ;  ft  Pegasi,  17°.15  ; 
a  Andromedse,  1°.09.  All  the  hour-angles  are  greater  than  those  of 
y  Pegasi,  so  that  all  the  stars  precede  y  Pegasi.  By  using  all  the 
observations  we  may  presumably  obtain  more  accurate  results,  and 
it  will  be  well,  as  in  all  cases  when  a  considerable  number  of 
observations  must  be  dealt  with,  to  arrange  the  reductions  in  a 
more  systematic  manner. 

In  the  table  on  the  following  page  the  difference  of  hour-angle  is 
obtained  by  subtracting  the  observed  hour-angle  in  each  case  from 
the  hour-angle  of  y  Pegasi,  so  that  its  value  is  negative,  if,  as  in 
the  results  given  above,  the  stars  precede  y  Pegasi,  and  positive 
if  they  follow  it.  An  observation  of  Venus,  'made  on  the  same 
occasion,  is  added  to  the  list,  and  an  additional  observation  of 
a  Pegasi  is  included ;  the  erroneous  observation  of  y  Pegasi  at 
7h  21m  30s  is  excluded. 

The  values  of  the  hour-angle  of  y  Pegasi  at  the  successive  times, 
as  given  in  column  6,  are  computed  from  the  following  considera- 
tions, the  proof  of  which  is  left  to  the  student.  If  a  quantity 
changes  uniformly,  and  its  values  at  several  different  times  are 
known,  the  mean  of  these  values  is  the  same  as  the  value  which 


LABORATORY    ASTRONOMY 


STAB 

TIME 

DECL. 

H.A. 

H.A.  OF  y  PEG. 

STAB  FOLLOWS 
y  PEG. 

1  Venus 

7h  12m  0s 

+  4°.0 

7  5°.  5 

65°.  86 

-  9°.64 

2  a  Peg. 

13  0 

+  15  .1 

83  .1 

66  .10 

-  17  .00 

3  7  Peg. 

14  0 

+  15  .2 

66  .3 

66  .35 

+  0  .05 

4  a  Peg. 

15  0 

+  15  .2 

83  .6 

66  .59 

-  17  .01 

5  /3  Peg. 

16  15 

+  28  .1 

84  .1 

66  .89 

-  17  .21 

6  a  Androm. 

17  10 

+  29  .0 

68  .3 

67  .12 

-  1  .18 

7  7  Peg. 

18  30 

+  15  .2 

67  .6 

67  .44 

.16 

8  a  Androm. 

22  30 

+  29  .1 

69  .6 

68  .44 

-  1  .16 

9  |8  Peg. 

23  30 

+  28  .1 

85  .9 

68  .88 

-  17  .22 

10  a  Peg. 

24  30 

+  15  .3 

86  .0 

68  .93 

-  17  .07 

11  7  Peg. 

25  30 

+  15  .1 

69  .1 

69  .17 

12  7  Peg. 

27  30 

+  15  .1 

69  .6 

69  .65 

+  0  .05 

the  quantity  has  at  the  mean  of  the  times.     Using  this  principle, 
we  find  the  hour-angle  of  y  Pegasi  at  7h  21ra  22s  was  68°.15. 

Between  observations  3  and  12  it  changed  3°.3  in  13Jm,  or  0°.244 
per  minute.  Assuming  this  rate  of  change,  it  is  easy,  though  labori- 
ous, to  compute  the  hour-angle  at  any  one  of  the  given  times ;  for 
example,  at  7h  12m  0s  the  hour-angle  was  68°.15  -  (9|§  times  0°.244), 
or  65°.86.  Labor  will  be  saved  by  making  a  table  of  the  values  at 
the  even  minutes  by  successive  additions  of  0°.244,  from  which  the 
values  at  the  observed  times  are  rapidly  interpolated.  The  sixth 
column  contains  the  number  of  degrees  by  which  the  hour-circle  of 
the  star  follows  that  of  y  Pegasi.  The  mean  values  for  each  star 
obtained  from  this  column  are  as  follows. 


STAB 

DECL. 

DIFF.  H.A. 

.7  Pegasi   
Venus  ........          . 

+  15°.  15 
-    4  .0 

o°.oo 

-    9  .64 

a  Pegasi 

+  15    20 

—  17  .03 

j8  Pegasi    
a  Androm. 

+  28  .10 
+  29  .05 

-  17  .22 
1  .17 

The  true  values  of  the  declinations  of  these  stars  as  determined  by 
many  years  of  observations  are  for  y  Pegasi  14°. 63,  a  Pegasi  14°.67, 
ft  Pegasi  27°.55,  a  AndromedsB  28°.53.  The  values  from  our 


THE    COMPLETE    SPHERE    OF    THE    HEAVENS 


53 


observations  are  15°.15,  15°.20,  28°.10,  29°.05,  so  that  the  latter 
require  corrections  of  —  0°.52,  —  0°.53,  —  0°.55,  and  —  0°.52,  respec- 
tively. This  is  due  to  a  faulty  adjustment  of  the  instrument,  but 
the  error  from  this  cause  evidently  affects  all  the  observations  by 
nearly  the  same  amount,  0°.53,  so  that  the  relative  positions  are 
given  quite  accurately  ;  our  observations  placing  the  whole  constel- 
lation about  y  too  far  north. 

Since  Venus  is  in  the  near 
neighborhood  of  y  Pegasi,  we 
may  assume  that  the  observa- 
tions of  that  planet  are  subject 
to  the  same  corrections,  that 
she  preceded  y  Pegasi  by  9°.64, 
and  that  her  true  declination 
was  -  4°.0  -  0°.53,  or  -  4°.53. 
The  correction  is  applied  alge- 
braically with  the  same  sign  as 
to  the  other  stars,  since  it  must 
be  so  applied  as  to  make  the 
true  place  farther  south  than 
the  observed  place. 

The  places  of  the  Square  of  Pegasus  and  the  planet  Venus,  as 
seen  in  the  sky  Feb.  5,  1900,  are  shown  in  Fig.  35. 

Before  plotting  the  stars  on  the  hemisphere  from  the  above 
data,  it  must  be  prepared  by  drawing  upon  it  in  their  proper  posi- 
tions the  meridian,  zenith,  pole,  and  equator.  Draw  the  hour-circle 
of  y  Pegasi  (see  Fig.  19,  p.  17)  at  the  proper  hour-angle  from  the 
meridian,  to  give  its  position  at  the  time  of  the  last  observation, 
which  may  be  determined  by  making  it  intersect  the  equator  at  the 
proper  point  69°.  6  west  of  the  meridian,  and  place  the  star  upon  it 
at  a  distance  from  the  equator  equal  to  the  observed  declination, 
15°.  14.  The  hour-angle  of  a  Pegasi  should  be  drawn  in  the  same 
manner  to  cut  the  equator  at  8  6°. 66  from  the  meridian,  and  the 
star  placed  upon  it  at  the  observed  declination,  15°. 20.  Of  course 
on  the  scale  of  so  small  a  hemisphere  the  nearest  half  degree  is 
sufficiently  accurate.  Bern  ember  that  the  configurations  on  the 
hemisphere  and  on  the  map  are  semi-inverted. 


QfAndrom. 
fega, 

rtfj 

/ 

« 

Venus 

D°                         -15°                       -30° 
FIG.  35 

54  LABORATORY    ASTRONOMY 

CLOCK   REGULATED  TO  SHOW  THE    HOUR-ANGLE  OF  THE 
FUNDAMENTAL   STAR 

The  method  of  calculating  the  hour-angles  of  y  Pegasi  in  the  last 
example  shows  that  if  the  reading  of  the  watch  can  be  relied  upon, 
the  observations  of  that  star  need  only  be  made  at  the  beginning 
and  at  the  end  of  the  period  of  observation,  the  hour-angle  at  any 
time  being  determined  by  its  uniform  increase ;  or  even  from  a 
single  observation  at  the  beginning  of  the  period,  since  at  the  time 
of  observation  of  any  star  the  hour-angle  of  y  Pegasi  can  be 
inferred  from  that  at  its  first  observation  by  adding  the  number  of 
degrees  which  it  would  have  described  in  the  time  elapsed,  obtained 
by  multiplying  the  number  of  hours  by  15,  or,  what  gives  the  same 
results,  dividing  the  minutes  by  4.  Moreover,  if  the  rate  of  the 
watch  is  such  that  it  completes  its  24  hours  in  the  time  in  which 
the  stars  complete  their  daily  revolution,  and  if  its  hands  are  so  set 
as  to  read  12  hours  when  y  Pegasi  is  on  the  meridian,  the  difference 
of  hour-angle  at  any  time  will  be  equal  to  the  reading  taken  directly 
from  the  hands  of  the  watch  reduced  as  above  to  degrees,  for  when 
the  star  is  on  the  meridian  and  its  hour-angle  therefore  zero,  the 
watch  marks  Oh  Om  0s.  Four  minutes  later  by  the  watch  the  hour- 
angle  of  the  star  has  increased  by  the  diurnal  revolution  to  1°; 
in  four  minutes  more  to  2° ;  when  the  watch  indicates  1  hour,  the 
star's  hour-angle  has  increased  to  15°,  and  so  on,  till  24  hours  have 
elapsed,  when  the  star  will  again  be  on  the  meridian  and  the  cycle 
recommences. 

The  rate  of  an  ordinary  watch  is  sufficiently  near  to  that  of  the 
stars  to  allow  of  its  use  for  this  purpose  for  periods  of  an  hour 
without  causing  any  error  in  our  observations. 

In  the  use  of  this  method  we  may  regard  the  observation  of  the 
fundamental  or  zero  star  as  a  means  of  finding  out  whether  the 
clock  is  set  to  the  right  time :  thus,  in  the  following  set  of  obser- 
vations the  first  observation  gives  the  hour-angle  of  y  Pegasi  67°. 6 
at  7h  15m  10s,  but  as  67°. 6  equals  4h  30m  24s,  we  may  regard  the 
clock  as  2h  44m  46s  fast ;  and  by  applying  this  correction  to  all  the 
observed  times,  may  write  down  at  once  under  the  title  "  corrected 
time  "  what  the  readings  would  have  been  if  the  clock  had  been  set 


THE    COMPLETE    SPHERE    OF    THE    HEAVENS 


55 


to  show  0  hours,  when  the  star's  hour-angle  was  0°.     Multiplying 
these  by  15  we  have  the  hour-angle  in  degrees  given  in  column  4. 

The  following  observations  were  undertaken  for  determining  the 
configuration  of  the  stars  in  Orion  and  its  neighborhood,  Feb.  6, 1900. 


STAR 

OBS.  TIME 

CORRECTED 
TIME 

H.A.  OF 
y  PEG. 

OBSERVED 
H.A.  OF  STAR 

DECL. 

FOLLOWS 
y  PEG. 

7  Pegasi 

7h  15m  IQs 

4h  3()m  24« 

67°.  6 

67°.  6 

+  15°.5 

a 

7  20   0 

4  35  14 

68.8 

348  .5 

-  1  .4 

80°.  3 

b 

7  22   0 

4  37  14 

69  .3 

349  .95 

-  0  .6 

79  .35 

c 

7  23  30 

4  38  44 

69  .7 

348.1 

-  2  .2 

81  .8 

d 

7  25  20 

4  40  34 

70.1 

344  .8 

+  7  .1 

85  .3 

e 

7  27  10 

4  42  24 

70  .6 

353  .0 

+  6.1 

77  .6 

f 

7  28  45 

4  43  59 

71  .0 

347  .5 

-  9  .9 

83  .5 

g 

7  30  20 

4  45  34 

71  .4 

356  .4 

-  8  .2 

75  .0 

h 

7  32   0 

4  47  14 

71  .8 

351  .6 

-  5  .5 

80  .2 

i 

7  34   0 

4  47  14 

72  .3 

334  .7 

-16  .9 

97  .6 

j 

7  35  45 

4  50  59 

72  .7 

321  .3 

+  5  .05 

111  .4 

a 

7  37  45 

4  52  59 

73.2 

352  .9 

-  1  .4 

80  .3 

7  Pegasi 

7  39  50 

4  55   4 

73  .8 

73.9 

+  15  .4 

Moon 

7  42   0 

4  57  14 

74  .3 

27  .6 

+  20  .4 

46.7 

The  results  of  columns  6  and  7  enable  us  to  map  the  constellation 
as  in  Fig.  36. 

One  or  two  constellations  may  be  plotted  in  this  manner  both  on 
the  map,  which  shows  the  constellation  as  seen  in  the  sky,  and  on 


,  j  A" 

jjftgasi 

•Prt 

cyon 

• 

Orion 

,  in" 

* 

"20* 

•  Sir 

ius 

105°            90*             75'            60°           45°            30°             15°             C 
FIG.  36. 

56  LABORATORY    ASTRONOMY 

the  hemisphere,  where  it  is  semi-inverted.  It  will  be  advisable, 
however,  before  much  work  has  been  done  in  this  way,  to  introduce 
a  slight  modification. 

THE   VERNAL   EQUINOX— RIGHT   ASCENSION 

The  precession  of  the  equinoxes  causes  a  change  in  the  position 
of  the  equator,  which  slowly  changes  the  declinations  of  all  the 
stars.  For  this  reason  it  is  found  more  convenient  to  select,  instead 
of  y  Pegasi  as  a  zero  star,  the  point  upon  the  equator  at  which  the 
sun  crosses  it  from  south  to  north  about  March  21  of  each  year. 
This  point,  which  is  called  the  vernal  equinox,  is  not  fixed,  but 
its  motion,  due  to  precession,  is  simpler  than  that  of  any  star  which 
might  be  selected  as  a  zero  point ;  it  precedes  the  hour-circle  of 
y  Pegasi  at  present  by  about  8  minutes  of  time,  or  2°  of  arc,  and  it 
was  because  of  this  proximity  that  we  first  selected  that  star. 

Instead,  therefore,  of  adjusting  our  clock  so  that  it  reads  Oh  Om  0s 
when  y  Pegasi  is  on  the  meridian,  we  set  it  to  that  time  when  the 
vernal  equinox  is  in  that  plane ;  its  readings  then  give  the  hour-angle 
of  the  vernal  equinox,  and  the  difference  between  the  hour-angles  of 
that  point  and  of  the  star  may  be  directly  obtained  from  our  obser- 
vations. The  distance  by  which  a  star  follows  the  vernal  equinox 
is  called  its  right  ascension;  more  carefully  defined,  it  is  the  arc 
of  the  equator  intercepted  between  the  hour-circle  of  the  star  and 
the  hour-circle  of  the  vernal  equinox  (which  measures  the  wedge 
angle  between  the  planes  of  these  circles) ;  it  is  also  the  angle  between 
the  tangents  drawn  to  these  two  circles  where  they  intersect  at  the 
pole.  Since  any  star  which  is  east  of  the  vernal  equinox  follows  it, 
the  right  ascensions  of  different  stars  increase  toward  the  east,  that 
is,  toward  the  left  in  the  sky  as  we  face  south,  but  toward  the  right 
on  the  solid  hemisphere  as  we  look  down  from  the  outside  upon  its 
southern  face. 

Hereafter  we  shall  fix  the  positions  of  the  stars  by  their  right 
ascensions  and  declinations.  We  may  make  use  of  the  observations 
already  reduced  with  very  little  additional  labor.  Since  y  Pegasi 
follows  the  vernal  equinox  by  2°,  we  need  only  add  that  amount  to 
the  quantities  given  in  column  7  on  page  55  to  know  the  right 


THE    COMPLETE    SPHERE    OF    THE    HEAVENS  57 


ascension  of  the  different  stars.  If  we  learn  later  that  on  Febru- 
ary 6  the  right  ascension  of  y  Pegasi  was  more  exactly  Oh  8m  58.64, 
we  may  further  correct  by  adding  5s,  or  even  58.64,  if  the  accuracy 
of  the  observations  warrants  it.  The  method  of  determining  the 
exact  position  of  the  zero  star  with  reference  to  the  vernal  equinox 
is  given  in  Chapter  VI. 

Formerly  right  ascensions  were  measured  altogether  in  degrees, 
but  owing  to  the  modern  use  of  clocks,  it  has  long  been  customary 
to  give  them  in  hours  ;  for  this  reason  the  hour-circle  of  instruments 
mounted  as  equatorials  is  graduated  to  read  hours  and  minutes 
directly.  Since  our  universal  equatorial  is  intended  to  serve  also 
as  an  altazimuth,  its  circles  are  both  graduated  to  degrees. 

SIDEREAL   TIME 

In  the  last  section  right  ascension  has  been  defined  as  the  angle 
between  the  hour-circle  passing  through  a  star  and  the  great  circle 
passing  through  the  pole  and  the  vernal  equinox.  The  latter  circle 
is  called  the  equinoctial  colure.  We  have  also  suggested  the  use 
of  a  clock  set  to  read  Oh  Om  0s  at  the  time  when  the  vernal  equinox 
is  on  the  meridian ;  so  that  the  hour-angle  of  the  vernal  equinox  at 
any  time  will  be  given  directly  by  the  reading  of  the  face  of  the 
watch  in  hours,  minutes,  and  seconds,  from  which  the  angle  in 
degrees  is  found  by  multiplying  by  15.  A  clock  set  in  this  manner, 
and  running  at  such  a  rate  that  it  completes  24  hours  in  the  time 
that  the  star  completes  its  revolution  from  any  given  hour-angle  to 
the  same  hour-angle  again,  is  said  to  keep  sidereal  time.  We 
shall  find  later  that  a  clock  so  regulated  gains  about  4  minutes  a 
day  on  a  clock  keeping  mean  time,  thus  gaining  24  hours  on  an 
ordinary  clock  in  the  course  of  a  year,  and  agreeing  evidently  with 
a  clock  keeping  apparent  time,  as  defined  on  page  19,  at  that  time 
when  the  sun  is  at  the  vernal  equinox  and  crosses  the  meridian  at 
the  same  time  with  the  latter. 

Let  us  suppose  now  that  the  vernal  equinox  has  passed  the 
meridian  by  one  hour,  then  its  hour-angle  is  lh,  or  15°;  and  our 
sidereal  clock  indicates  exactly  lh  Om  0s.  Any  star  which  is  at  this 
time  on  the  meridian,  that  is,  whose  hour-angle  is  0°,  must  therefore 


58 


LABORATORY    ASTRONOMY 


follow  the  vernal  equinox  by  lh,  or  15°,  while  at  the  same  instant 
the  time  by  our  sidereal  clock  is  lh  Om  0s.  By  our  definition  of 
right  ascension,  since  the  star  follows  the  vernal  equinox  by  lh,  its 
right  ascension  is  lh ;  in  this  case,  therefore,  the  right  ascension  of 
the  star  in  hours,  minutes,  and  seconds  has  the  same  value  as  the 
time  given  by  the  hands  of  the  clock.  In  the  same  way,  if  the 
vernal  equinox  has  passed  the  meridian  so  far  that  its  hour-angle  is 
2h  15m,  the  face  of  the  clock  will  show  2h  15m  j  and  any  star  then 
upon  the  meridian  follows  the  vernal  equinox  by  2h  15m.  The  same 
relation  holds  here ;  namely,  that  the  right  ascension  of  the  star  is 
equal  to  the  time  by  the  sidereal  clock  when  the  star  is  upon  the 
meridian.  This  might  have  been  given  as  a  definition  of  the  term 
"  right  ascension  " ;  and,  indeed,  so  closely  are  the  two  connected 
in  the  mind  of  the  practical  astronomer  that  if  the  right  ascension 
of  a  star  is  given,  he  at  once  thinks  of  this  number  as  representing 
the  time  of  its  meridian  passage. 


RIGHT    ASCENSION    PLUS    HOUR-ANGLE    EQUALS 
SIDEREAL   TIME 

We  may  here  give  an  explanation  of  a  general  principle  of  very 
frequent  application,  and  of  which  this  is  simply  a  particular  case. 
Suppose  the  vernal  equinox,  represented  by  the  symbol  T  (Fig.  37),  to 

have  passed  the  meridian  by  5h  10m. 
Then  a  star,  S,  whose  right  ascen- 
sion is  2h  15m,  since  it  follows  the 
vernal  equinox  by  that  amount, 
will  have  passed  the  meridian  by 
2h  55m ;  and  its  hour-angle  will  be 
2h  55m.  The  arc  of  the  equator 
between  the  meridian  and  the  ver- 
nal equinox  may  be  considered  as 
made  up  of  two  parts :  the  right 

ascension  of  the  star,  which  is  measured  by  the  arc  eastward 
from  the  vernal  equinox  to  the  hour-circle  of  the  star,  and  the 
hour-angle  of  the  star,  which  extends  from  the  meridian  westward 
to  the  hour-circle  of  the  star.  Since  this  is  true  of  any  star,  or, 


FIG.  37 


THE    COMPLETE    SPHERE    OF    THE    HEAVENS  59 

indeed,  of  any  heavenly  body,  we  may  make  the  following  general 
statement :  The  right  ascension  of  any  body  plus  its  hour-angle  at 
any  instant  will  be  equal  to  the  sidereal  time  at  that  instant ;  or, 
as  it  is  sometimes  written :  R.A.  +  H.A.  =  Sidereal  Time.  If  the 
body  is  a  point  on  the  meridian,  its  H.A.  =  zero ;  hence  the  E/.A. 
of  a  star  on  the  meridian,  or  briefly,  R.A.  of  the  meridian  =  Sidereal 
Time,  as  we  have  before  shown.  « 

From  this  relation  we  may  most  simply  determine  the  right 
ascension  of  any  heavenly  body  by  observing  its  hour-angle  with 
the  equatorial  instrument,  and  at  the  same  time  noting  the  sidereal 
time,  since  R.A.  =  Sidereal  Time  —  H.A.  It  is  by  this  method 
that  we  shall  now  proceed  to  make  a  somewhat  extended  catalogue 
of  stars  from  which  we  may  plot  their  positions  upon  the  globe. 

We  will  here  notice  some  of  the  important  uses  to  which  this 
principle  may  be  put.  If  by  any  other  means  the  right  ascension 
of  a  body  is  known,  we  may  find  its  hour-angle  at  any  given  sidereal 
time  by  the  equation,  Sidereal  Time  —  R.A.  =  H.A.  This  gives  us 
an  easy  way  to  point  upon  any  object  whose  right  ascension  and 
declination  are  known,  if  we  have  a  clock  keeping  sidereal  time ; 
and  this  is  the  usual  way  in  which  the  astronomer  finds  the  objects 
which  he  wishes  to  observe,  since  they  are  generally  so  faint  that 
they  cannot  be  seen  by  the  naked  eye.  For  example,  to  point  the  tele- 
scope at  the  great  nebula  in  Orion,  whose  right  ascension  is  5h  28m, 
and  declination  6°  S.,  we  first  set  the  decimation  circle  to  —  6°, 
and  if  the  sidereal  time  is  7h  30m  we  set  the  hour-circle  to  2h  2m, 
then  the  telescope  will  be  pointed  upon  the  star.  If  the  sidereal 
time  is  4h  30m,  in  which  case  the  star  evidently  has  not  reached 
the  meridian  by  nearly  an  hour,  we  must  add  24  hours  to  the  sidereal 
time ;  then  the  expression,  H.A.  =  Sidereal  Time  —  R.A.  will 
become  H.A.  =  28h  30m  —  5h  28m,  or  23h  2m,  the  hour-angle  being 
reckoned,  as  before  stated,  from  Oh  up  to  24h.  If  then  the  hour- 
circle  is  brought  to  the  reading  345£°  =  15°  x  23^,  we  shall  find 
the  star  in  the  field. 


60  LABORATORY    ASTRONOMY 


THE   CLOCK   CORRECTION 

The  same  principle  enables  us  to  set  our  clock  correctly  to  sidereal 
time  by  observing  the  hour-angle  of  any  star  whose  right  ascension 
is  known.  For  example,  the  right  ascension  of  Sirius  being  6h  40™, 
or  100°,  and  its  hour-angle  being  observed  to  be  330°,  or  22h,  the 
sidereal  time  is  R.A.  -f  H.A.,  that  is,  430°,  or,  subtracting  360°,  is 
70°,  corresponding  to  4h  40m;  and  a  clock  may  be  set  to  agree;  or, 
by  subtracting  the  time  which  it  then  indicates,  we  determine  a 
correction  to  be  applied  to  its  reading  to  give  the  true  sidereal 
time.  If,  for  instance,  at  the  observation  above,  the  clock  time  is 
4h  41m  10s,  the  clock  correction  is  —  lm  108.  In  this  case  the  clock 
is  lm  10s  fast,  the  time  which  it  indicates  is  greater  than  the  true 
time,  and  its  "error"  is  said  to  be  -f-  lm  10s.  On  the  other  hand, 
when  the  clock  is  slow  the  correction  to  true  time  is  positive,  while 
the  "  error  "  is  negative. 

It  is  customary  to  observe  this  distinction  between  the  terms 
"  error  "  and  "  correction  "  ;  the  former  is  the  amount  by  which  the 
observed  value  of  a  quantity  exceeds  its  true  value,  while  the  correc- 
tion is  the  quantity  which  must  be  added  to  the  observed  to  obtain  the 
true  value.  They  are  thus  numerically  equal  but  of  opposite  sign. 

The  error  of  the  declination  circle  determined  by  the  observations 
of  page  53  was  -f-  0°.53,  while  the  correction  was  —  0°.53. 

For  the  constantly  occurring  "  clock  correction,"  we  shall  use  the 
symbol  A£,  the  value  of  which  is  positive  if  the  clock  is  slow  and 
negative  if  it  is  fast. 

If,  as  is  often  desirable,  we  wish  to  observe  a  body  of  known  right 
ascension  upon  the  meridian,  we  have  only  to  observe  it  when  the 
time  by  the  sidereal  clock  is  equal  to  its  right  ascension. 

We  may  of  course  find  the  right  ascension  of  the  moon  by  a  direct 
comparison  with  the  neighboring  stars,  just  as  we  have  determined 
the  difference  in  right  ascension  of  a  Pegasi,  from  that  of  y  Pegasi, 
for  the  brighter  stars  can  be  easily  observed  at  the  same  time  as  the 
moon ;  but  no  star  is  so  bright  that  it  can  be  readily  observed  by 
our  small  instrument  when  the  sun  is  above  the  horizon,*  and  we 
have  therefore  no  means  of  making  a  direct  comparison  between 

*  See,  however,  page  69. 


THE    COMPLETE    SPHERE    OF    THE    HEAVENS  61 

a  star  and  the  sun.  But  by  means  of  our  clock  and  our  new  method 
of  observation  this  becomes  easy ;  and  the  sun  is  to  be  added  to  the 
list  of  bodies  whose  right  ascension  we  are  to  observe  regularly.  It 
is  only  necessary  that  we  should  be  provided  with  a  clock  which 
keeps  correct  sidereal  time.  (See  page  67.) 

We  have  already  spoken  of  the  means  of  setting  the  clock; 
now  a  few  words  as  to  how  the  regularity  of  its  rate  may  be  deter- 
mined. It  is  only  necessary  to  observe  the  watch  time  at  which 
any  star  is  at  a  given  hour-angle  on  successive  nights.  If  the 
rate  of  the  clock  is  such  that  the  interval  between  the  observa- 
tions is  greater  than  24  hours,  the  watch  is  gaining ;  if  the  amount 
is  less  than  half  a  minute  a  day,  the  watch  may  be  assumed  for  our 
purposes  to  be  keeping  correct  sidereal  time,  its  actual  error  at  any 
time  being  checked,  as  before  described,  by  the  observation  of  the 
hour-angle  of  bodies  of  known  right  ascension. 

LIST   OF   STARS 

Our  first  care  will  be  to  observe  a  number  of  bright  stars  not  very 
far  from  the  equator  which  will  serve  for  setting  the  clock  or  deter- 
mining its  error,  selecting  them  so  that  several  shall  always  be  above 
the  horizon  and  may  at  any  time  be  used  for  this  purpose.  Several 
of  those  already  observed  will  be  found  in  the  list  given  on  the 
following  page,  which  contains  the  approximate  places  of  a  number 
of  conspicuous  stars. 

By  repeated  comparisons  of  these  stars  with  each  other  and  with 
y  Pegasi,  their  right  ascensions  may  easily  be  fixed  within  308,  and 
they  may  then  be  used  for  determining  the  clock  error  at  any  time 
when  they  are  visible.  The  observations  of  each  evening  should  be 
reduced  as  soon  as  possible  and  maps  made  of  the  various  constella- 
tions similar  to  those  of  Figs.  35  and  36 ;  it  is,  however,  impossible 
to  represent  any  large  portion  of  the  sphere  satisfactorily  on  a 
plane  surface,  and,  in  order  to  have  a  proper  idea  of  the  relative 
positions  of  the  various  constellations,  we  must  plot  our  results  on 
a  globe  —  a  proceeding  still  more  necessary  when  we  come  to  study 
the  motion  of  the  sun  and  moon  among  the  stars  by  the  method  of 
the  following  chapter. 


62 


LABORATORY    ASTRONOMY 


A  globe  6  inches  in  diameter  is  sufficiently  large  for  our  purpose ; 
it.  should  be  so  mounted  that  it  may  be  turned  about  its  axis  011  a 
firm  support,  and  upon  it  should  be  traced  24  hour-circles  15°  apart, 
and  small  circles  (parallels  of  declination)  parallel  to  the  equator 
and  10°  apart ;  its  surface  should  be  smooth  and  white,  and  of  such 
a  texture  as  to  take  a  lead-pencfl  mark  easily,  but  permit  of  erasure. 


TIME    STARS 


STAB 

MAG. 

K.A. 

6 

STAB 

MAG. 

R.A. 

8 

7  Pegasi 

3 

Oh.l 

+  15° 

Denebola 

2 

llh.7 

+  15° 

/S  Ceti 

2 

0  .6 

-  19 

52  Corvi 

3 

12  .4 

-  16 

/3  Andromedse 

2 

1  .1 

+  35 

Spica 

1 

13  .3 

-  11 

a  Arietis 

2 

2  .0 

+  23 

Arcturus 

1 

14  .2 

+  20 

a  Ceti 

2£ 

3  .0 

+    4 

a2  Librae 

3 

14  .8 

+  16 

Alcyone 

3 

3  .7 

+  24 

a  Serpentis 

3 

15  .7 

+    7 

Aldebaran 

1 

4  .5 

+  16 

Autares 

1 

16  .4 

-26 

Capella 

1 

5  .2 

+  45 

a  Ophiuchi 

2 

17  .5 

+  13 

Rigel 

1 

5  .2 

-    8 

72  Sagittarii 

3 

18  .0 

-  30 

e  Orionis 

2 

5  .5 

1 

Vega 

1 

18  .6 

+  39 

Betelgeuze 

1 

5  .8 

+    7 

Altai  r 

1 

19  .8 

+    9 

Sirius 

1 

6  .7 

-  17 

a2  Capricorni 

4 

20  .2 

-  13 

Castor 

2 

7  .5 

+  32 

a  Delphini 

4 

20  .6 

+  16 

Procyon 

1 

7  .6 

+    5 

e  Pegasi 

H 

21  .7 

+    9 

Pollux 

1 

7  .7 

+  28 

a  Aquarii 

3 

22  .0 

-    1 

a  Hydrae 

2 

9  .4 

-    8 

a  Pegasi 

2i 

23  .0 

+  15 

Regulus 

1 

10  .1 

+  12 

The  number  attached  to  the  Greek  letter  indicates  that   the   star  to  be 
observed  is  the  following  of  two  neighboring  stars. 


CHAPTER   V 
MOTION   OF   THE   MOON   AND   SUN   AMONG  THE   STARS 

FOR  plotting  the  stars  on  the  globe  in  their  proper  places,  as  given 
by  their  right  ascensions  and  declinations,  it  is  convenient  to  have 
the  equator  graduated  into  spaces  of  10m  each  ;  this  may  be  done 
by  laying  the  edge  of  a  piece  of  paper  along  the  equator,  and  mark- 
ing off  the  points  of  intersection  of  the  equator  with  two  consecu- 
tive hour-circles  ;  laying  the  paper  upon  a  flat  surface,  bisect  the 
space  between  the  two  lines  with  the  dividers,  and  trisect  each  of 
these  spaces  by  trial,  testing  the  equality  of  the  spacing  by  the 
dividers  ;  this  may  be  satis- 
factorily done  by  two  or  three 
trials,  and  the  short  scale  thus 
obtained  may  be  easily  trans- 
ferred to  the  arcs  on  the  equator 
between  each  two  hour-circles. 
It  may  be  found  convenient  to 
bisect  each  of  the  spaces  on  the 
scale,  thus  dividing  the  equator 
into  spaces  of  5m  each. 

A  strip  of  parchment  or 
parchment  paper  about  8  inches 
long  and  £  inch  wide,  of  the 
shape  shown  in  Fig.  38,  and 
graduated  to  degrees,  completes 
the  apparatus  necessary  for 
plotting.  The  hole  being 
placed  over  the  axis  of  the 
globe,  the  graduated  edge  of 

the  strip  may  be  made  to  coincide  with  the  hour-circle  of  any  star 
by  causing  it  to  intersect  the  equator  at  a  point  corresponding  to  the 
star's  right  ascension,  taking  care  that  the  edge  lies  in  a  great  circle 

63 


FlG  38 


64  LABORATOKY    ASTRONOMY 

of  the  sphere ;  the  graduated  edge  gives  at  once  the  proper  declina- 
tion for  plotting  the  star  upon  its  hour-circle,  and  the  point  may  be 
marked  with  a  well-sharpened,  hard  lead  pencil ;  the  latter  should 
be  carefully  kept,  and  used  for  purposes  of  plotting  only.  With 
this  simple  apparatus  the  stars  may  be  rapidly  and  accurately 
placed  upon  the  globe. 

An  attempt  should  be  made  to  represent  the  magnitudes  of  the 
stars  by  the  size  of  the  dots  which  indicate  their  places. 

THE  MOON'S  PATH  ON  THE  SPHERE 

The  moon  should  be  placed  on  the  list  of  objects  for  regular 
observation,  the  observations  being  made  in  precisely  the  same 
manner  as  those  of  the  stars,  and  its  place  should  be  plotted 
upon  the  globe  at  each  observation  and  marked  by  a  number, 
giving,  the  date  of  the  month.  This  method  of  fixing  the  moon's 
place  is  much  more  accurate  than  those  made  use  of  in  Chapter  II, 
and,  as  the  places  are  plotted  upon  a  globe,  we  may  study -to  better 
advantage  those  peculiarities  of  her  motion  which  are  masked  by 
the  distortion  of  the  map  referred  to  in  Chapter  II. 

The  position  of  the  node  may  now  be  fixed  with  such  a  degree  of 
accuracy  that  its  regression  is  shown  by  the  observations  of  two  or 
three  months,  if  some  care  is  taken  to  observe  as  nearly  as  possible  at 
the  same  altitude  in  the  successive  months,  so  that  the  corrections 
for  parallax  may  be  nearly  the  same ;  indeed,  a  very  few  months  will 
force  upon  the  notice  of  the  observer  the  fact  that  the  moon's  path 
does  not  lie  in  one  plane,  just  as  observations  a  few  days  apart 
show  that  the  sun's  diurnal  path  is  not  really  a  small  circle  lying- 
in  one  plane. 

We  also  study  the  variable  motion  of  the  moon  by  applying 
dividers  between  the  successive  plotted  places  and  then  placing 
the  dividers  against  the  parchment  scale  to  measure  the  distance 
in  degrees  traversed  in  the  plane  of  the  orbit.  The  scale  must 
lie  along  an  hour-circle  so  as  to  conform  to  the  curvature  of  the 
sphere. 

The  average  rate  being  about  13°  a  day,  the  points  on  the  orbit 
should  be  determined  as  nearly  as  possible  at  which  the  motion  is 


MOTION  OF  THE  MOON  AND  SUN  AMONG  THE  STARS      65 


greater  and  less  than  this  amount,  and  the  point  of  most  rapid 
motion  fixed  as  closely  as  possible ;  this  point  is  most  simply  fixed 
by  its  distance  an  degrees  from  the  ascending  node  of  the  moon's 
orbit.  Since  the  latter  point,  however,  is  continually  changing, 
it  is  customary  to  reckon  the  so-called  "  longitude  in  the  orbit "  of 
the  point  by  measuring  from  the  vernal  equinox  along  the  ecliptic 
to  the  node,  and  adding  the  angle  measured  along  the  orbit  from 
the  node  to  the  point. 

The  variations  of  the  moon's  angular  diameter  and  the  point  of 
the  orbit  where  the  diameter  is  greatest  should  be  compared  with 

V777          VH  Vf  V 


-20 


-50 


120 


110 


100 


70  60  50 

FIG.  39 


30  20 


the -results  obtained  from  the  investigation  of  the  angular  velocity 
in  the  orbit,  since  we  thus  gain  some  knowledge  of  the  moon's 
relative  distances  from  us  at  different  points  of  its  orbit,  and  of  the 
relation  between  its  distance  and  its  rate  of  motion  about  the  earth. 
The  scale  of  the  6-inch  globe  is  too  small  to  do  justice  to  the 
accuracy  of  our  observations,  which  are  accurate  to  a  quarter  or  a 
tenth  of  a  degree,  and  it  will  be  interesting  to  plot  these  observations 


66  LABORATORY    ASTRONOMY 

on  a  map  constructed  on  a  larger  scale,  and  on  a  plan  which 
reduces  the  distortion  to  very  'small  limits  in  the  region  of  the 
ecliptic ;  such  a  map  is  shown  in  Fig.  39  on  a  reduced  scale ;  the 
ecliptic  is  here  taken  as  a  straight  horizontal  line,  as  the  equator 
is  in  the  star  map  previously  used  ;  the  latitude,  or  angular  distance 
of  a  point  from  the  ecliptic  measured  on  a  great  circle  perpendicular 
to  the  latter,  serves  as  the  coordinate  corresponding  to  the  declina- 
tion on  our  former  map,  while  right  ascension  is  replaced  by  lon- 
gitude, or  distance  along  the  ecliptic  measured  from  the  vernal 
equinox  up  to  360°.  The  same  map  will  serve  also  for  plotting 
the  paths  of  the  planets  in  our  later  study. 

For  convenience  in  plotting,  the  parallels  of  declination  and  the 
hour-circles  are  printed  in  broken  lines  upon  the  map.  The  obser- 
vations of  the  moon  shown  in  the  figure  are  those  of  December, 
1899,  already  plotted  on  the  map  of  Fig.  25. 

THE    SUN'S   PLACE   AMONG   THE   STARS 

By  means  of  the  equatorial  we  may  also  determine  the  place  of 
the  sun  among  the  stars,  although  the  method  of  direct  comparison 
with  stars  we  have  used  in  the  case  of  the  moon  is  not  applicable, 
since  the  stars  are  not  visible  when  the  sun  is  above  the  horizon ; 
the  most  obvious  method  which  is  capable  of  any  degree  of  accuracy 
involves  the  use  of  a  clock  regulated  to  sidereal  time. 

To  determine  the  place  of  the  sun,  point  upon  it  with  the  equa- 
torial about  two  hours  before  sunset ;  note  the  time,  and  read  the 
circles  ;  as  soon  as  possible  after  sunset  observe  a  star  in  the  same 
manner,  with  the  instrument  as  near  as  may  be  to  its  position  at 
the  sun  observation.  It  is  evident  that  if  the  circumstances  were 
fortunately  such  that  the  telescope  did  not  have  to  be  moved  between 
the  observations,  the  difference  in  right  ascension  of  the  sun  and  the 
star  would  be  the  difference  in  time  noted  by  the  sidereal  clock, 
while  the  declinations  of  the  sun  and  star  would  be  the  same.  The 
nearer  the  star  is  to  the  position  in  which  the  sun  was  observed,, 
the  less  will  be  the  errors  arising  from  imperfect  adjustment  and 
orientation  of  the  instrument;  while  the  shorter  the  interval  be- 
tween the  observations,  the  smaller  will  be  the  error  due  to  the 


MOTION  OF  THE  MOON  AND  SUN  AMONG  THE   STARS      67 

uncertainty  in  the  rate  of  the  clock.  As  the  condition  of  not 
moving  the  telescope  can  seldom  be  fulfilled,  however,  we  must 
treat  the  observation  as  follows : 

Let  E.A.,  H.A.,  t,  and  A£  be  the  right  ascension,  hour-angle, 
clock  time,  and  clock  correction  at  the  time  of  the  star  observation, 
and  E.A.',  H.A.',  t',  and  A£,  the  corresponding  quantities  at  the 
sun  observation.  The  equation 

E.A.  +  H.A.  =  Sidereal  Time  =  t  -f  A£ 

determines  the  value  of  A£,  which  substituted  in  the  equation 
Sidereal  Time  =  t1  +  A£  ==  E.A.'  +  H.A.' 

determines  the  value  of  E.A.',  the  sun's  right  ascension  at  the 
moment  of  observation. 

The  value  of  A#,  as  determined  from  the  first  equation,  will  be 
negative  if  the  clock  is  fast,  and  positive  if  the  clock  is  slow ;  and 
it  must  always  be  applied  to  the  observed  time  with  the  proper 
sign.  The  declination  of  the  sun  is,  of  course,  given  directly  by 
the  reading  of  the  declination  circle. 

The  following  example  illustrates  the  method : 

March  29,  1899,  an  observation  of  the  sun  with  an  equatorial 
telescope,  and  a  clock  keeping  sidereal  time,  gave  the  following 
values : 

Observed  time  =  5h  36m  26s ;  H.A.  =  75°.7  =  5h  2m  488 ;  3  =  +  4°.l. 
About  an  hour  after  sunset  an  observation  of  a  Ceti  was  made 
in  nearly  the  same  position  of  the  instrument,  which  gave  the 
following  values  : 

Observed  time  =  7h  53m  43s ;  H.A.  =  74°.l  =  4h  56m  24s ;  3  =  +  4°.2. 
This  latter  gives,  from  the  known  right  ascension  of  a  Ceti, 

2h  57m  Q.  +  4h  56m  24B  =  Sidereal  Time  =  7h  53m  43s  +  A*, 

and  hence  A£  =  —  19s ;  and,  applying  the  same  equation  to  the-  sun 
observation, 

Sun's  E.A.  +  5h  2m  48s  =  5h  36m  26s  -  19s  =  5h  36m  7s ; 

hence  the  sun's  right  ascension  at  the  time  of  the  first  observation 
was  Oh  33m  19s.  This  is  liable  to  an  error  equal  to  the  uncertainty  of 
the  circle  readings,  which  may  be  at  least  one-twentieth  of  a  degree, 


68  LABORATORY    ASTRONOMY 

or  12s  of  time,  and  to  an  error  equal  to  the  uncertainty  of  the  gain 
or  loss  of  the  clock  during  the  interval  of  2^-  hours  between  the 
two  observations,  probably  five  or  ten  seconds  of  time.  We  may 
assume  that  the  errors  arising  from  defective  adjustment  of  the 
instrument  were  the  same  for  both  objects,  and  may  be  neglected, 
since  the  position  of  the  instrument  was  very  nearly  the  same  for 
both  observations. 

DIFFERENTIAL   OBSERVATIONS 

The  declination  of  a  Ceti,  as  read  from  the  circles,  was  +4°.2, 
while  its  known  decimation  was  -f  3°.  7.  The  correction  necessary 
to  reduce  the  circle  reading  to  the  true  value  is,  therefore,  —  0°.5, 
and,  applying  this  quantity  to  the  reading  on  the  sun,  we  have  for 
the  true  value  of  the  sun's  declination  +  4°.l  —  0°.5  =  -f  3°. 6.  It  is 
worthy  of  note  that  the  correction  is  about  the  same  as  that  deter- 
mined from  the  observations  discussed  on  page  53,  which  were 
made  with  the  same  instrument  in  nearly  the  same  adjustment,  but 
from  a  different  place  of  observation.  These  results  indicate  an 
inherent  defect  in  the  instrument,  which  is  at  least  in  great  part 
neutralized  by  the  method  of  observation.  It  is  a  very  important 
thing,  even  with  the  most  delicate  instruments,  to  avail  ourselves  of 
methods  which  accomplish  this  object,  and  surprisingly  good  work 
may  be  done  with  poor  instruments  by  paying  proper  attention 
to  the  details  of  observation  for  this  purpose. 

Methods  by  which  an  unknown  body  is  thus  compared  with  a 
known  body  under  circumstances  as  nearly  alike  as  possible  are 
called  "differential  methods.7' 

INDIRECT    COMPARISON   OF    THE    SUN   WITH    STARS 

It  is  often  possible  to  determine  the  difference  of  right  ascension 
of  the  sun  and  some  well-known  star  by  using  the  moon  as  an  inter- 
mediary, determining  the  difference  of  right  ascension  of  the  sun  and 
moon  during  the  daytime  and  comparing  the  moon  and  a  star  as  soon 
as  possible  after  sunset,  the  motion  of  the  moon  during  the  interval 
being  allowed  for.  The  irregularity  of  the  moon's  motion  may, 


MOTION  OF  THE  MOON  AND  SUN  AMONG  THE   STARS      69 

however,  introduce  a  greater  error  than  that  arising  from  uncertainty 
in  the  rate  of  the  clock.  A  better  method  is  offered  on  those  not 
infrequent  occasions  when  the  planet  Venus  is  at  its  greatest 
brilliancy,  when  it  may  be  easily  observed  in  full  daylight ;  the 
motion  of  Venus  in  the  interval  is  much  smaller  and  more  nearly 
uniform,  and,  therefore,  more  accurately  determined ;  and  by  this 
method  the  interval  between  the  observations  connecting  the  sun 
with  Venus  and  Venus  with  the  star  may  be  reduced  to  a  very  few 
minutes,  or  even  seconds,  so  that  the  error  due  to  the  clock  may  be 
regarded  as  negligible. 

The  following  observations  illustrate  the  method. 


1900 

WATCH  TIME 

H.A. 

6 

April  19.3. 

Procyon      

8h  17m45s 

15°  4 

+    5°  5 

Venus    ..... 

8   19    33 

56   1 

+  26   1 

Procyon      .     .     .     ... 
Venus    »    .  -    .     .     .     .     .  .- 
Procyon 

8   21    45 
8   23      0 
8   24    53 

16  .4 

57  .0 
17   2 

+    5  .55 
+  26  .05 

+    53 

April  20.0. 

Sun    .     . 

1    28    45 

358   5 

+  116 

Venus    ... 

1   31    35 

313   2 

+  25   35 

Sun   . 

1   36    10 

0    15 

+  11    6 

Venus    
Sun                                  .     . 

1   38    33 
1   41    21 

315  .0 
1    4 

+  25.3 
+  116 

April  20.  3. 

Procyon      .... 

9  31    27 

33   25 

+    5   45 

Venus 

9  32    30 

72    9 

+  25    0 

Procyon      
Venus    ...          ... 

9   33    28 
9  35      0 

33.9 

+    5  .4 
+  25   9 

Procyon      

9   36      0 

34  .25 

+    5.4 

The  observations  April  19.3,  that  is,  April  19  about  7  P.M.,  give  for 
the  hour-angle  of  Venus  5  6°.  55  at  the  watch  time  8h  21m  17s,  and  for 
that  of  Procyon  16°.33  at  Sh  21m  28s;  hence  at  8h  21m  Procyon 
followed  Venus  40°.22. 

In  the  same  way  we  find  that  April  20.3  Procyon  followed  Venus 
39°.3,  the  change  of  the  right  ascension  of  Venus  being  0°.92  in  25.2 
hours.  A  simple  interpolation  shows  that  April  20.0  Procyon 


TO  LABORATORY    ASTRONOMY 

followed  Venus  39°.59,  and  the  observations  at  that  time  show  that 
Venus  followed  the  sun  45°.92,  so  that  Procyon  followed  the  sun 
39°.59  +  45°. 92  =  85°.51,  and  the  difference  of  right  ascension  between 
Procyon  and  the  sun  at  noon  on  April  20  was,  therefore,  5h  42m  2s. 

ADVANTAGES    OF    THE   EQUATORIAL   INSTRUMENT 

Observation  with, the  equatorial  we  shall  find  especially  useful 
in  getting  exact  positions  of  the  moon,  since  it  is  available  at  any 
time  when  the  moon  is  above  the  horizon,  and  after  sunset  we  can 
always  find  some  bright  star  sufficiently  near  to  afford  a  fairly 
accurate  value  of  its  place. 

It  is  often  inconvenient  to  observe  the  moon  by  the  more  accurate 
method  which  is  described  in  Chapter  VI,  that  of  meridian  observa- 
tions, which  is  confined  to  a  short  interval  of  one  or  two  minutes 
each  day,  and  is  often  interfered  with  by  clouds  passing  at  the 
critical  moment,  although  nine-tenths  of  the  whole  day  may  be 
suitable  for  observations  made  out  of  the  meridian.  Moreover, 
until  the  moon  is  several  days  old,  it  is  too  faint  for  observation  at 
its  meridian  passage.  It  is,  therefore,  upon  the  equatorial  that  we 
shall  mainly  rely  for  the  determination  of  the  moon's  motion,  as 
well  as  for  many  observations  of  the  planets  out  of  the  meridian. 

Although  it  is  far  more  convenient  to  find  the  right  ascension  and 
declination  of  the  sun  by  the  method  of  the  following  chapter,  at 
least  a  few  positions  should  be  found  by  observations  with  the 
equatorial  and  plotted  on  the  globe.  The  result  will  be  to  show 
that  the  path  of  the  sun  is  very  exactly  a  great  circle  fixed  on  the 
sphere  or  so  nearly  fixed  that  some  years  of  observation  with  the 
most  refined  instruments  are  necessary  to  detect  any  change  in  its 
position  among  the  stars,  although  a  much  shorter  time  even  would 
serve  to  show  the  slow  change  of  its  intersection  with  the  celestial 
equator  due  to  precession. 

This  great  circle  is  called  the  ecliptic,  and  its  position  is  shown  on 
the  map  which  we  have  used  for  plotting  our  first  moon  observation. 

Three  months  will  give  a  sufficient  arc  of  this  circle  to  enable  us 
to  determine  with  some  accuracy  its  position  with  respect  to  the 
equator,  its  inclination  to  the  latter,  and  their  points  of  intersection ; 


MOTION  OF  THE  MOON  AND  SUN  AMONG  THE   STAKS      71 

if  possible,  observations  should,  however,  be  continued  throughout 
the  year  which  the  sun  requires  to  complete  its  circuit,  so  that  the 
variability  of  its  motion  may  be  observed,  most  of  the  work,  how- 
ever, being  done  with  the  meridian  circle. 

The  sun's  diameter  should  occasionally  be  measured  to  determine 
the  points  at  which  it  is  nearest  to  and  farthest  from  the  earth. 


CHAPTER   VI 
MERIDIAN   OBSERVATIONS 

WE  have  now  arrived  at  a  point  where  we  can  see  what  are  the 
desirable  conditions  for  making  observations  as  accurately  as  possible 
of  the  position  of  a  heavenly  body.  To  adjust  the  equatorial  instru- 
ment so  that  its  axis  lies  in  the  meridian  and  at  the  proper  inclina- 
tion, and  to  keep  it  so  adjusted,  is  a  matter  of  some  difficulty.  In 
the  last  chapter  we  have  shown  how,  by  observing  an  unknown  body 
in  a  certain  fixed  position  of  the  instrument,  and  later  a  body  whose 
right  ascension  and  declination  are  known  in  as  nearly  as  possible 
the  same  position  of  the  instrument,  we  lessen  the  effect  of  the 
instrumental  errors.  We  made  our  observation  of  the  sun  shortly 
before  sunset,  so  that  the  interval  between  this  observation  and  that 
of  the  comparison  star  should  be  as  short  as  possible.  If,  however, 
the  rate  of  the  clock  can  be  relied  upon,  there  is  no  reason  why  the 
observation  should  not  be  made  when  the  sun  is  on  the  meridian, 
the  interval  of  time  required  to  connect  it  with  stars  in  that  case 
being  not  necessarily  more  than  eight  or  nine  hours  in  the  most 
extreme  case ;  and  the  comparative  ease  with  which  an  instrument 
may  be  constructed  so  that  it  shall  be  at  all  observations  exactly  in 
the  meridian,  and  the  possibility  of  constructing  very  accurate  time- 
pieces, has  determined  the  use  of  such  instruments  for  all  the  more 
precise  observations  in  astronomy,  such  as  fix  the  positions  of  the 
fundamental  stars  and  the  vernal  equinox  on  the  celestial  sphere. 

The  equatorial  instrument  may  be  used  for  this  purpose  by  clamp- 
ing it  in  such  a  position  that  the  reading  of  the  hour-circle  is  0°,  in 
which  case  the  declination  axis  is  horizontal  east  and  west,  and 
when  the  telescope  is  moved  about  its  axis  it  always  lies  in  the 
plane  of  the  meridian.  If,  with  the  instrument  so  adjusted,  we 
observe  the  sun  at  the  time  of  its  meridian  passage,  we  may  find 
its  declination  by  reading  the  declination  circle,  and  its  right  ascen- 
sion by  noting  the  interval  which  elapses  before  the  meridian  transit 

72 


MERIDIAN    OBSERVATIONS 


73 


of  some  known  star  after  nightfall,  free  from  any  error  involved  in 
reading  the  hour-circle.  As  before,  a  star  should  be  chosen  at 
nearly  the  same  declination,  so  that  the  interval  of  time  may  be 
very  nearly  equal  to  the  difference  in  right  ascension  between  the 
sun  and  the  star,  even  if  the  instrument  is  not  very  exactly  in  the 
meridian.  Observation  of  several  different  stars  will  enable  us  to 
determine  whether  the  instrument  actually  does  describe  the  plane 
of  the  meridian  as  it  is  rotated  about  the  horizontal  axis  (see  Chapter 
VIII) ;  and  by  the  observation  of  stars  near  the  pole,  as  described 
on  page  81,  we  may  determine  whether  the  declination  circle  reads 
exactly  0°  when  the  telescope  points  to  the  equator,  as  should  be 
the  case. 

THE   MERIDIAN   CIRCLE 

An  instrument  which  is  to  be  used  in  this  manner,  however,  is 
not  usually  so  constructed  that  it  can  be  pointed  at  any  point  in  the 
heavens.  Thus,  it  is  un- 
necessary that  it  should 
consist  of  so  many  moving 
parts  as  the  equatorial  in- 
strument, and  steadiness, 
strength,  and  ease  of  ma- 
nipulation are  very  much 
increased  by  constructing 
it  as  shown  in  Fig.  40, 
which  represents  a  very 
small  instrument  built  on 
the  plan  of  the  meridian 
circle  of  the  fixed  observ- 
atory. The  strong  hori- 
zontal axis  revolves  in 
two  Y's,  which  are  set 
in  strong  supports  in  an 
east  and  west  line.  The 
axis  is  enlarged  towards 
the  center,  and  through  the  center  passes  at  right  angles  the 
telescope  tube.  The  axis  carries  at  one  end  a  graduated  circle 


FIG.  40 


74  LABORATOKY   ASTRONOMY 

perpendicular  to  the  axis  of  rotation.  If  the  axis  of  the  telescope 
is  perpendicular  to  the  axis  of  rotation,  and  if  the  latter  is  adjusted 
horizontally  east  and  west,  the  telescope  may  be  brought  into  any 
position  of  the  meridian  plane,  but  must  always  be  directed  to  some 
point  of  the  latter.  A  pointer  attached  to  the  support  marks  the 
zero  of  the  vertical  circle  when  the  telescope  points  to  the  zenith, 
and  if  the  telescope  be  pointed  to  a  star  at  the  time  of  its  meridian 
passage,  the  angle  as  read  off  on  the  circle  is  the  zenith  distance  of 
the  star ;  while  the  time  of  the  star's  meridian  passage  by  a  clock 
giving  true  sidereal  time  is  its  right  ascension.  If  the  latitude  of 
the  place  of  observation  is  known,  the  star's  decimation  is  deter- 
mined by  the  fact  that  the  zenith  distance  plus  the  declination  of 
any  body  equals  the  latitude  (see  page  81).  At  first  the  latitude 
may  be  used  as  determined  by  the  sun  observation  of  Chapter  I,  or 
from  a  good  map  showing  the  place  of  observation,  but  ultimately 
its  value  should  be  determined  with  the  meridian  circle  itself. 


LEVEL   ADJUSTMENT 

We  will  now  proceed  to  show  how  to  make  the  necessary  adjust- 
ments for  placing  the  telescope  so  that  it  may  move  in  the  plane  of 
the  meridian. 

Place  the  instrument  on  its  pier  and  bring  the  Y's  as  nearly  as 
possible  into  an  east  and  west  line.  If  the  pier  is  the  same  that 
has  been  used  in  the  previous  work,  this  may  be  done  by  bringing 
the  telescope  into  the  meridian  which  has  been  determined  by  the 
method  of  equal  altitudes. 

The  axis  must  first  be  brought  into  a  horizontal  line,  making  use 
for  this  purpose  of  the  striding  level  (Fig.  41),  which  is  a  necessary 
auxiliary  of  this  instrument.  This  is  a  glass  tube  nearly  but  not 
quite  cylindrical,  ground  inside  to  such  a  shape  that  a  plane  passing 
through  its  axis,  CD,  cuts  the  wall  in  an  arc,  AB,  of  a  circle  whose 
center  is  at  0.  In  this  tube  is  hermetically  sealed  a  very  mobile 
liquid  in  sufficient  quantity  nearly  but  not  quite  to  fill  it  —  the 
space  remaining,  called  the  "  bubble,"  always  occupying  the  top 
of  the  tube.  When  CD  is  horizontal,  the  bubble  rests  in  the 
middle  of  the  tube  with  its  ends,  of  course,  at  equal  distances  from 


MERIDIAK    OBSERVATIONS 


75 


the  middle;  the  tube  is  graduated  so  that  this  distance  may  be 
measured,  the  numbering  of  the  graduations  usually  increasing  in 
both  directions  from  the  center  of  the  tube.  If  the  radius  of  the 
arc  is  14.3  feet,  a  length  of  3  inches  of  this  arc  will  be  equal  to 

about  1°,  since  the  arc  of  1°  in  any  circle  is  about of  the  radius ; 

o7.o 

1  inch  of  the  arc  will  then  be  about  20',  and  0.05  inch  1'.  These 
are  about  the  actual  values  for  the  level  used  with  the  instrument 


\1 

\i 


FIG.  41 


shown  in  Fig.  40,  the  scale  divisions  being  about  fa  of  an  inch 
apart  and  therefore  corresponding  to  an  arc  of  1'. 

If  the  line  CD  is  inclined  at  an  angle  of  1'  to  the  horizontal 
line  by  raising  the  end  A,  the  center  of  curvature  will  be  dis- 
placed toward  the  left,  and  the  level  will  have  the  same  inclina- 
tion as  if  the  whole  tube  had  been  turned  to  the  right  about  the 
point  0  through  an  angle  of  1';  and  the  highest  point  of  the  arc, 
which  is  always  directly  above  0,  is  now  fa  of  an  inch  from  the 
middle  toward  A.  Since  the  bubble  always  rests  at  the  highest  point 
of  the  arc,  it  follows  that  its  ends  will  each  be  moved  toward  A  by 
one  division ;  if,  for  instance,  the  readings  of  the  ends  are  5  and  5 
when  CD  is  horizontal,  they  will  be  6  and  4  when  CD  is  inclined 


76  LABORATORY   ASTRONOMY 

by  1',  and  evidently  7  and  3  when  CD  is  inclined  2',  etc.,  the  incli- 
nation in  minutes  of  arc  being  one-half  the  difference  of  the  readings 

of  the  ends   of   the  bubble,  or  — - —  if  A   and  B  represent  the 

2i 

readings  of  the  ends  of  the  bubble  in  each  case.  If  the  reading  of 
B  is  greater,  the  end  A  is  depressed  by  one-half  the  difference  of 
the  readings ;  and  the  above  expression  applies  to  both  cases  if  we 
agree  that  it  shall  always  denote  the  elevation  of  A,  a  negative  value 

of  — - —  indicating  depression  of  A. 


REVERSAL   OF   THE  LEVEL 

The  level  tube  is  attached  to  a  frame  (Fig.  40)  resting  on  two  stiff 
legs  terminating  in  Y  's,  which  are  of  the  same  shape  and  size  as 
those  in  which  the  axis  of  the  meridian  circle  rests,  the  axis  of  the 
level  tube  being  adjusted  as  nearly  as  possible  parallel  to  the  line 
joining  the  Y's.  It  is  difficult  to  insure  this  condition,  but  if  it  is 
not  exactly  fulfilled,  the  horizontality  of  the  axis  may  still  be  deter- 
mined by  placing  the  level  on  the  axis,  and  determining  the  value 

— - — ,  and  then  turning  it  end  for  end,  and  again  reading  the  value ; 

L 

for  if  the  end  A  is  high  by  the  same  amount  in  each  case,  the  axis 
is  obviously  horizontal,  and  the  measured  angle  of  inclination  is  due 
to  the  fact  that  the  leg  of  the  level  adjacent  to  A  is  longer  than  the 
other  leg.  The  practical  rule  is  to  read  the  west  and  east  ends  in 

TJ/-      T^ 

each  position.    If  these  readings  are  Wl  El  Wz  E2,  — ^  —  is  the 

£ 

elevation  of  the  west  end  according  to  the  first  observation,  and 

TT7"       -Cl 

— 2-jr — -  at  the   second.      If  the  leg  which  is  west  at  the  first 

£ 

observation  is  too  long,  the  first  observation  gives  a  value  for  the 
elevation  of  the  west  end  too  great,  and  the  second  a  value  too 
small  by  the  same  amount ;  and  the  average  of  the  two  values 

TTT-      77  Tl^     rj 

— ^ — -  and  — 2-jij — -  gives  the  true  value  of  the  inclination  of  the 
axis. 


MERIDIAN    OBSERVATIONS  77 


It  is  usual  to  write  this   (w*+w*\    (Ei  +  E*)  and  to  record 

4 
the  observations  in  the  following  form : 

W1  E, 


Subtract  the  second  sum  from  the  first  and  divide  by  4.  This 
gives  a  positive  value  if  the  west  end  is  high,  and  the  axis  may 
be  made  horizontal  by  turning  the  leveling  screw  so  as  to  make 
the  level  bubble  move  through  the  proper  number  of  divisions. 
The  level  should  be  again  determined  in  the  same  way,  and  the 
axis  is  level  when 


The  following  record  of  level  observation  made  Feb.  26.3,  1900, 
conforms  to  the  above  scheme : 

W  E 

1*  2* 


-1 

-  \  division  =  15" 

The  west  end  being  too  low,  the  screw  was  turned  so  as  to  raise  it 
enough  to  move  the  bubble  J  division  toward  the  west,  the  level 
remaining  on  the  axis  during  the  adjustment  and  watched  as  the 
screw  was  turned ;  the  readings  were  then  as  follows : 

2i  If 

If  2J 

4  4 

0 

And  the  axis  was  truly  level,  since  (  Wi  4-  TF2)  —  (El  -{-  Ez)  =  0. 


1 


78  LABORATORY   ASTRONOMY 


COLLIMATION   ADJUSTMENT 

The  line  of  collimation  of  the  telescope  is  the  line  drawn  from 
the  center  of  the  lens  to  the  wires  that  cross  in  the  center  of  the 
field.  When  the  telescope  is  "  pointed  "  or  "  set "  upon  a  star,  the 
image  of  the  star  falls  upon  the  point  where  these  wires  cross,  and 
when  the  instrument  is  correctly  adjusted  the  line  of  collimation  is 
perpendicular  to  the  axis  of  rotation,  so  that  the  line  of  collimation 
cuts  the  celestial  sphere  in  a  great  circle  as  the  telescope  turns  upon 
its  axis. 

To  make  this  adjustment,  point  the  telescope  exactly  upon  any 
well-defined  distant  point,  —  the  meridian  mark  will,  of  course,  be 
chosen  if  it  has  been  located,  —  then  remove  the  axis  from  its  Y's 
and  replace  it  after  turning  it  end  for  end ;  if  the  telescope  is  still 
set  on  the  mark  in  the  second  position,  the  adjustment  is  correct; 
otherwise  move  the  wire  halfway  toward  the  mark  by  means  of 
the  screws  a,  a  (Fig.  40).  Set  again  upon  the  mark  by  moving 
the  screws  in  the  eyepiece  tube ;  reverse  the  axis  again,  and  thus 
continue  until  the  telescope  points  exactly  upon  the  mark  in  both 
positions  of  the  axis. 

If  the  adjustments  for  level  and  collimation  are  properly  made, 
the  intersection  of  the  wires  in  the  center  of  the  field  of  view  will 
appear  to  describe  a  vertical  circle,  that  is,  a  great  circle  through 
the  zenith,  as  the  instrument  is  turned  on  its  axis.  The  final 
adjustment  consists  in  bringing  this  circle  to  coincide  with  the 
meridian,  but  for  this  we  must  have  recourse  to  observations  of 
stars. 

» 

AZIMUTH   ADJUSTMENT 

The  simplest  method  is  to  observe  the  time  of  transit  by  a  sidereal 
clock  of  a  circumpolar  star  at  its  upper  transit,  and  again,  12  hours 
later,  at  its  transit  below  the  pole ;  if  the  interval  is  exactly  12 
hours,  the  adjustment  is  correct ;  if  the  interval  is  less  than  12 
hours,  the  telescope  evidently  points  west  of  the  pole,  and  the  west 
end  of  the  rotation  axis  must  be  moved  toward  the  north.  This  is 
done  by  the  screws  a,  a  (Fig.  40),  the  fraction  of  a  turn  being  noted  j 


MERIDIAN    OBSERVATIONS  79 

the  observation  is  repeated  upon  the  following  night,  and  by  com- 
paring the  change  which  has  been  produced  by  moving  the  screws, 
the  further  alteration  required  is  readily  estimated.  On  Feb.  26, 
1900,  the  lower  transit  of  e  Ursse  Minoris  was  observed  at  4h  58m 
12s,  and  the  upper  transit  at  16h  53m  32s;  the  times  were  taken  by 
a  sidereal  clock  and  have  been  corrected  for  its  error ;  the  interval 
being  llh  55m  40s,  it  is  evident  that  the  telescope  pointed  to  the 
east  of  the  meridian,  the  arc  of  the  star's  diurnal  path  between 
the  lower  and  upper  transits  lying  to  the  east  of  the  meridian  and 
being  less  than  12h  by  4m  20s  or  260s. 

To  correct  the  error,  the  west  end  of  the  axis  was  moved  toward 
the  south  by  turning  the  adjusting  screws  through  one-quarter  of 
a  turn.  On  the  following  day  the  observations  were  repeated  as 
follows : 

Feb.  27.25,  lower  transit  4h  54m  45s ;  Feb.  27.75,  upper  transit 
16h  54m  28s ;  the  eastern  arc  was  still  too  small,  but  the  error  had 
been  reduced  to  17&,  and  required  a  further  correction  of  ^?75  of  a 
quarter  turn  of  the  screws,  which  were  therefore  turned  through 
about  6°  in  the  same  direction  as  before,  and  the  instrument  was 
thus  brought  very  closely  into  the  meridian. 

This  method  can  only  be  used  with  small  instruments  when  the 
night  is  more  than  12  hours  long ;  but  it  is  the  only  independent 
method ;  it  requires  that  the  rate  of  the  clock  shall  be  known 
between  the  two  observations,  and  it  requires  observations  at  in- 
convenient times.  A  more  convenient  method  is  always  used  in 
practice,  but  requires  an  accurate  knowledge  of  the  right  ascensions 
of  a  considerable  number  of  stars  in  the  neighborhood  of  the  pole. 

It  has  been  stated  that  it  is  often  inconvenient  to  observe  the  moon 
when  on  the  meridian,  but  with  this  exception  all  the  fundamental 
observations  of  astronomy  are  now  made  with  meridian  instruments 
on  account  of  the  simplicity  and  permanence  of  the  necessary  adjust- 
ments. A  body  observed  on  the  ineridian  is  also  at  its  greatest 
altitude  and  least  affected  by  atmospheric  disturbances,  which  often 
interfere  with  the  observation  of  bodies  near  the  horizon. 


80  LABORATORY   ASTRONOMY 

DETERMINATION   OF   DECLINATIONS   WITH   THE 
MERIDIAN   CIRCLE 

The  circle  of  the  meridian  instrument  may  be  used  to  determine 
the  declination  of  a  star  in  two  ways,  of  which  that  now  described 
is  perhaps  the  most  obvious,  but  also  the  least  convenient. 

If  the  reading  of  the  circle  is  known  when  the  telescope  is  pointed 
at  the  pole,  the  angle  through  which  the  telescope  must  be  moved  to 
point  upon  any  star,  that  is,  the  polar  distance  of  the  star,  is  the 
difference  between  this  value  and  the  circle  reading  when  the  tele- 
scope is  pointed  at  the  star ;  this  angle  is  90° — the  star's  declination; 
if  the  star  is  on  the  equator,  the  angle  is  90°;  and  if  the  star  is 
south  of  the  equator,  the  angle  is  greater  than  90°  by  an  amount 
equal  to  the  declination  of  the  star ;  if  we  consider  the  declination  a 
negative  quantity  for  a  star  south  of  the  equator,  the  value  90°  —  8 
represents  the  polar  distance  in  all  cases. 

To  determine  the  reading  of  the  "polar  point"  we  may  set  the 
telescope  upon  a  circumpolar  star  at  its  "upper  culmination"  and 
read  the  circle,  and  again,  12  hours  later,  set  on  the  same  star  at  its 
"  lower  culmination/'  the  mean  of  the  two  readings  is  the  reading 
of  the  polar  point.  The  effect  of  refraction  may  be  neglected  with 
our  small  instruments  without  causing  an  error  of  J^  of  a  degree 
at  any  place  in  the  United  States  if  we  restrict  ourselves  to  stars 
within  10°  of  the  pole,  or  the  circle  readings  may  be  corrected  by 
a  refraction  table.  Immediately  after  making  this  determination  it 
is  advisable  to  make  a  setting  on  the  meridian  mark  and  note  the 
reading ;  this  point  may  thereafter  be  used  as  a  reference  point  from 
which  the  reading  of  the  polar  point  may  be  at  any  time  determined 
if  the  meridian  mark  has  not  in  the  mean  time  changed  its  position. 

Better  still,  the  observation  of  the  polar  point  may  be  combined 
with  a  determination  of  the  circle  reading  when  the  telescope  points 
at  the  zenith,  by  one  of  the  methods  to  be  described  later;  the 
difference  of  the  readings  in  this  case  is  obviously  equal  to  the 
co-latitude,  and  such  an  observation  constitutes  an  "absolute  deter- 
mination of  the  latitude,"  that  is,  a  determination  made  without 
reference  to  observations  made  at  any  other  place.  When  the  lati- 
tude has  once  been  satisfactorily  determined,  the  observations  of 


MERIDIAN    OBSERVATIONS 


81 


the  declinations  of  stars  can  be  made  to  depend  upon  determinations 
of  the  zenith  point  by  means  of  the  fact  that  for  a  body  on  the  meridian 

Declination  =  Latitude  —  Zenith  Distance, 

latitude  and  declinations  being  reckoned  positive  northward  from 
the  plane  of  the  equator,  and  zenith  distance  positive  southward 
from  the  zenith.  The  proof  of  this  relation  is  left  to  the  student 
as  well  as  the  interpretation  of  the  result  when  the  observation  is 
made  at  the  transit  below  the  pole. 

At  the  time  of  observing  the  transits  of  e  Ursse  Minoris  described 
on  page  79  the  following  readings  of  the  circle  were  made  when 
the  star  was  in  the  center  of  the  field.  Each  of  these  observations 
consists  of  two  readings :  one  of  the  index  A  on  the  south  end  of 
a  horizontal  bar  fixed  to  the  supports  of  the  axis,  and  the  other 
of  an  index  B  at  the  other  extremity  of  the  bar,  as  nearly  as 
possible  half  a  circumference  from  A.  An  angle  given  by  the 
mean  of  two  readings  made  in  this  manner  is  free  from  the  "  error 
of  eccentricity,"  which  affects  readings  by  a  single  index  in  case 
the  center  of  the  graduated  circle  does  not  exactly  coincide  with 
the  axis  about  which  it  is  turned  between  the  two  observations. 


DATE 

A 

B 

MEAN 

February  26.25    .     .     .     . 

55°.  45 

55°.  35 

55°.40 

26.75    .... 

39  .95 

39  .85 

39  .90 

27.25    .... 

55  .45 

55  .35 

55  .40 

27.75    .... 

39  .95 

39  .85 

39  .90 

Hence  the  reading  when  the  instrument  was  pointed  at  the  pole 


was 


55°.40  +  39°.90 


=  47°.65. 


Evidently  the  polar  distance  of  the  star  was 


55°  40  _  39°  90 

° 


—  7°-75, 


and  its  declination  82°.  25;  and  we  have  thus  obtained  an  "inde- 
pendent "  or  "  absolute  "  determination  of  the  declination  of  c  Ursae 
Minoris  ;  that  is,  a  determination  independent  of  the  work  of  other 
observers,  and  only  dependent  on  the  accuracy  of  our  circle  and  of 
our  observations. 


82  LABOR ATOKY    ASTRONOMY 

The  circle  was  known  to  be  adjusted  so  that  the  reading  of  the 
zenith  was  very  exactly  zero,  hence  the  latitude  of  the  place  of 
observation  was  42°.35.  The  exact  agreement  of  these  observations 
indicates  that  the  magnifying  power  of  the  telescope  was  such  that 
it  could  be  set  more  accurately  than  the  circle  could  be  read,  and 
not  that  the  results  are  reliable  to  a  hundredth  of  a  degree. 

For  convenience  in  recovering  the  zenith  reading,  in  case  the 
adjustment  of  the  circle  should  be  disturbed,  the  zenith  distance 
of  a  meridian  mark  was  measured  repeatedly,  the  result  showing 
that  its  polar  distance  was  137°. 47,  and  this  was  used  to  check  the 
polar  reading  in  later  observations  upon  stars  when  it  was  impossible 
to  get  observations  of  the  same  star  above  and  below  the  pole. 

Another  method  of  making  absolute  determinations  of  the  latitude 
with  the  meridian  circle  is  to  observe  the  zenith  distance  of  the  sun 
at  the  solstices  ;  the  mean  of  these  values  being  the  zenith  distance 
of  the  equator,  which  is  equal  to  the  latitude.  This  observation, 
however,  is  subject  to  considerable  uncertainty  on  account  of  the 
difference  in  atmospheric  conditions  at  the  summer  and  winter 
solstice,  and  to  great  inconvenience  on  account  of  the  lapse  of 
time ;  it  is,  however,  of  course,  the  means  upon  which  we  must 
rely  for  the  accurate  determination  of  the  obliquity  of  the  ecliptic, 
one  of  the  fundamental  quantities  of  astronomy. 

For  the  use  that  we  shall  make  of  the  meridian  circle,  it  will 
probably  be  most  convenient  to  make  a  careful  determination  of 
the  polar  distance  of  the  meridian  mark,  and  use  this  habitually 
as  a  point  for  reference. 

PROGRAM   OF   WORK   WITH    THE   MERIDIAN   CIRCLE 

Work  with  the  meridian  circle  should  at  first  consist  of  reobser- 
vation  of  all  the  stars  which  have  been  previously  observed  with 
the  equatorial,  except  those  which  are  west  of  the  meridian  after 
nightfall  and  cannot  be  observed  for  six  months.  Attention  should 
be  given  to  gathering  a  list  of  stars  within  15°  or  20°  of  the  pole 
for  the  purpose  of  quickly  setting  the  instrument  in  the  meridian 
by  the  methods  of  page  79.  The  sun  should  be  observed  at  least 
once  a  week  and  its  place  plotted  on  the  globe,  and  many  stars 


MERIDIAN    OBSERVATIONS  83 

in  the  neighborhood  of  the  moon's  path  to  form  a  basis  for  finding 
the  moon's  place  by  differential  observations,  of  course,  also  the 
moon  itself,  the  planets  and  a  comet,  if  any  of  sufficient  bright- 
ness appears.  In  this  way,  by  observing  a  few  stars  each  night, 
a  great  amount  of  material  may  be  stored  for  future  use. 

Especial  attention  should  be  given  to  getting  a  good  number  of 
observations  of  stars  near  the  equator,  so  that  fairly  accurate  values 
of  their  differences  of  right  ascension  may  be  obtained,  and  at  the 
first  opportunity  the  absolute  right  ascension  of  one  of  their  num- 
ber must  be  determined  in  order  that  thus  the  places  of  all  may  be 
known.  The  results  may  be  best  recorded  by  making  a  list  of 
their  right  ascensions  referred  to  an  assumed  vernal  equinox.  Thus, 
the  observations  discussed  on  page  52  show  that  a  Pegasi  precedes 
y  Pegasi  by  17°.03  =  lh  8m  7s,  or,  in  other  words,  follows  it  by 
22h  51m  53s ;  and  if  the  right  ascension  of  y  Pegasi  referred  to  the 
assumed  equinox  is  Oh  8m,  that  of  a  Pegasi  is  22h  59m  53s.  If 
in  the  course  of  the  year  observation  shows  that  the  true  right 
ascension  of  y  Pegasi  is  Oh  8m  5s,  it  is  evident  that  the  true  value 
for  a  Pegasi  is  22h  59m  58s,  and  that  the  right  ascension  of  all  stars 
referred  to  the  assumed  equinox  by  comparison  with  y  Pegasi  must 
be  increased  by  5s. 

DETERMINATION   OF    THE   EQUINOX 

An  opportunity  for  observing  the  absolute  right  ascension  of  the 
zero  star,  which  is  often  called  a  "  determination  of  the  equinox," 
occurs  about  the  middle  of  March  and  September. 

If  the  course  is  begun  in  September,  it  will  be  well  to  make  this 
determination  with  the  help  of  more  experienced  observers,  even 
before  the  nature  and  object  of  the  measures  are  understood. 

The  observation  consists  in  determining  the  difference  of  right 
ascension  of  some  star  from  the  sun  at  the  instant  when  the  latter 
crosses  the  equator,  for  at  that  time  it  is  either  at  the  vernal  or 
autumnal  equinox,  and  its  right  ascension  is  in  the  one  case  0  hours 
and  in  the  other  12  hours. 

If  a  meridian  observation  of  the  sun's  altitude  shows  that  the  sun 
is  exactly  on  the  equator  at  meridian  passage,  and  the  time  of  transit 


84  LABORATORY    ASTRONOMY 

is  noted  by  a  sidereal  clock,  and  as  soon  as  it  is  sufficiently  dark  the 
transit  of  a  star  is  observed,  the  difference  of  the  times  is  the  absolute 
right  ascension  of  the  star  if  the  observation  is  made  at  the  vernal 
equinox,  or  equals  the  right  ascension  of  the  star  minus  12h  if  the 
observation  is  made  at  the  autumnal  equinox. 

Inasmuch  as  the  meridian  of  the  observer  will  rarely  be  that 
one  on  which  the  sun  happens  to  be  as  it  crosses  the  equator,  we 
must  make  observations  on  the  day  before  and  the  day  after  the 
equinox,  thus  getting  the  difference  of  right  ascension  of  the  star 
from  the  sun  at  noon  on  both  days.  The  declination  of  the  sun 
being  also  measured  at  these  two  times,  a  simple  interpolation  gives 
the  time  at  which  the  sun  crossed  the  equator,  and  this  time  being 
known,  another  simple  interpolation  between  the  differences  of  right 
ascension  at  the  two  noons  gives  the  difference  of  right  ascension 
of  the  sun  and  star  at  the  time  when  the  sun  was  at  the  equinox, 
which  is  the  star's  absolute  right  ascension. 

The  first  interpolation  assumes  that  the  sun's  declination  changes 
uniformly  with  the  time,  and  the  second  that  its  right  ascension 
changes  uniformly  with  the  time. 

Observations  should  extend  over  a  period  of  a  week  before  and  a 
week  after  the  equinox  to  test  the  truth  of  these  assumptions. 

In  observing  the  sun,  a  shade  of  colored  or  smoked  glass  may  be 
placed  over  the  eyepiece,  or  the  eyepiece  may  be  drawn  out  as  in 
the  method  of  observation  described  on  page  37,  and  the  screen 
held  in  such  a  position  that  the  cross-wires  are  sharply  focused 
upon  it.  As  the  image  of  the  sun  enters  the  field  it  should  be 
adjusted  by  moving  the  telescope  slightly  north  or  south  till  the 
horizontal  wire  passes  through  the  center  of  the  disk,  and  as  the 
latter  advances,  the  time  should  be  noted  when  the  preceding  and 
following  limbs  cross  the  vertical  wire,  as  well  as  the  time  when 
the  vertical  wire  bisects  the  disk ;  at  the  instant  of  transit  the  disk 
should  be  neatly  divided  into  four  equal  divisions,  a  very  small 
deviation  from  this  condition  being  quite  perceptible  to  the  eye. 


MERIDIAK    OBSERVATIONS 


85 


THE   AUTUMNAL   EQUINOX   OF   1899 

The  following  table  gives  the  details  of  observations  taken  a*t 
the  autumnal  equinox  of  1899  for  the  purpose  of  determining  the 
equinox. 

The  latitude  of  the  place  of  observation  was  42°.5,  and  the  declina- 
tions given  in  the  last  column  are  calculated  by  subtracting  the  zenith 
distance  in  each  case  from  this  quantity,  as  explained  on  page  81. 


DATE 

OBJECT 

TIME  OF  TRANSIT 

ZEN.  DIST. 

DECL. 

Sept.  22 

Sun  

12h      Om    2s.O 

S42°.2 

+    0°.3 

77  Serpentis 

18      18      22.6 

45  .4 

-    2.9 

X  Sagittarii      .     . 

18      24        2.4 

67  .95 

-25.45 

Vega      .... 

18      35      44.5 

3  .87 

+  38  .63 

Altair    .... 

19      48        7.6 

33  .98 

+    8  .52 

Sept.  23 

Sun   . 

12       3      45.1 

42  .62 

—    0  .12 

17  Serpentis 

18      18      20.1 

45  .4 

-    2  .9 

X  Sagittarii     .     . 

18      23      57.3 

67  .97 

-  25  .47 

Vega      .... 

18      35      42.6 

3.85 

4-  38  .65 

Altair    .... 

19     48        1.5 

The  intervals  between  the  observed  times  of  transit  of  each  star 
on  the  two  different  dates  range  from  23h  59m  538.9  to  23h  59m  58M, 
showing  that  the  clock  was  losing  about  4s  daily,  a  quantity  so 
small  that  for  our  purpose  it  may  be  neglected. 

Observations  of  the  sun  made  on  different  dates  between  Sep- 
tember 18  and  September  23,  but  not  here  recorded,  showed  that 
its  right  ascension  and  declination  were  changing  uniformly  at  the 
rate  of  about  3m  45s  and  0°.39  per  day.  The  table  above  shows 
that  from  September  22  to  September  23  the  rates  were  3m  438.1 
(or,  allowing  for  clock  rate,  about  3m  39s)  and  0°.42  per  day,  and 
the  latter  value  we  shall  use  to  determine  the  time  of  the  equinox, 
as  follows  : 

At  noon  September  22,  or  September  22d.O,  as  it  is  expressed  by 
astronomers,  the  sun's  declination  was  +  0°.3,  and  September  23.0 


86 


LABORATORY   ASTRONOMY 


its  declination  was  —  0°.12.  Hence  its  declination  was  0°  Septem- 
ber 22f  f,  or  September  22d.714.  It  was  at  that  time,  as  exactly  as 
our  observations  can  show,  at  the  autumnal  equinox,  and  its  right 
ascension  was  12h  Om  0s. 

Since  rj  Serpentis  followed  it  to  the  meridian  6h  18m  208.6,  that 
quantity  is  the  difference  between  the  right  ascension  of  the  star 
and  that  of  the  sun  September  22.0.  Similarly  the  difference  of 
right  ascension  of  sun  and  star  September  23.0  was  6h  14m  35s.O ; 
that  is,  it  was  3m  458.6  less  than  at  the  previous  date.  Assuming 
this  change  to  be  uniform,  the  difference  of  right  ascension  of  sun 
and  star  at  the  moment  of  the  equinox  on  September  22d.714  was 
0.714  X  3m  458.6,  or  2m  41M  less  than  on  September  22.0;  that  is, 
it  was  6h  15m  39s. 5,  and  since  the  right  ascension  of  the  sun  Sep- 
tember 22.714  was  12h  Om  O8,  the  right  ascension  of  t]  Serpentis  was 
18h  15m  398.5. 

The  following  table  gives  the  data  from  which  the  "absolute 
right  ascensions  "  of  the  four  stars  are  thus  determined.  In  the 
last  column  are  the  declinations,  which  are  the  means  obtained  from 
several  observations  between  September  14  and  September  23. 


STAB 

R.A.  OF  STAB  MINUS  R.A.  OF  SUN 

STAB'S 
R.A.             DECL. 

SEPT.  22.0 

SEPT.  23.0 

SEPT.  22.714 

t\  Serpentis 
X  Sagittarii 
Vega 
Altair 

6*  18m  2(K6 
6    24      0.4 
6    35    42.5 
7    48      5.6 

6h  i4m  35s.  o 
6    20     12.2 
6    31     57.5 
7    44     16.4 

6*  15^  39«.5 
6    21     17.4 
6    33       1.8 
7    45     21  .9 

18h  15m  398.5 
18    21     17.4 
18    33       1.8 
19    45    21  .9 

-  2°.  89 
-25  .48 
+38  .65 
+  8  .59 

The  measurements  upon  which  the  above  results  depend  are  of 
two  kinds  :  observed  clock  times,  which  are  liable  to  errors  of  a 
very  few  seconds,  so  that  the  differences  of  right  ascension  may  be 
assumed  to  be  correct  within  perhaps  4s;  and  measures  of  the  sun's 
declination,  which  with  the  greatest  care  may  be  in  error  at  least 
0°.05  on  any  given  date. 

It  is  quite  wUhin  the  bounds  of  probability,  for  instance,  that 
the  sun's  declination  was  +  0°.25  on  September  22.0  and  —  0°.17 


MERIDIAN    OBSERVATIONS 


87 


on  September  23.0 ;  and  recomputing  with  these  values,  the  date 
of  the  equinox  was  September  22ff,  or  September  22d.595,  and 
the  right  ascensions  of  the  stars  18h  16m  68.4,  18h  21m  448.6, 
18h  33™  28S.6,  18h  45m  49S.2  ;  that  is,  the  uncertainty  of  the  equinox 
is  0.12  days  and  of  the  right  ascensions  about  278,  although  the 
relative  right  ascension  is  altered  only  by  a  fraction  of  a  second 
in  each  case.  It  is  thus  evident  that  the  accuracy  of  the  right 
ascensions  depends  chiefly  upon  the  accuracy  with  which  the  sun's 
declination  can  be  measured. 


THE  AUTUMNAL   EQUINOX   OF    1900 

In  order  to  increase  the  accuracy  of  determination  of  declination, 
a  new  circle  reading  to  minutes  of  arc  was  substituted  for  that 
used  for  the  observations  of  the  equinox  in  1899,  and  the  observa- 
tions were  repeated  at  the  same  place  in  1900.  The  weather  con- 
ditions were  unfavorable,  so  that  only  the  following  observations 
could  be  made. 


DATE 

OBJECT 

TIME  OF 

TRANSIT 

ZEN. 

DlST. 

DECL. 

Sept.  22 
Sept.  23 

Sun    - 
Vega                      .     . 

18 
19 

12 
19 

35 
47 

3 

48 

n    44s.  8 

27  .0 
49.0 

1  .5 
35.0 

842° 

0 

33 

42 
33 

ll'.S 
49.0 
51.0 

33.1 
54.0 

+    0° 
+  38 
+    8 

-    0 

+    8 

18'.  5 
41.0 
39.0 

3.1 
36.0 

Altair 

Sun    ...... 

Altair      .          ... 

From  these  data,  by  the  same  method  as  before,  the  date  of  the 
equinox  is  found  to  be  September  22^f;|,  or  September  22.8565. 
If  each  declination  of  the  sun  is  accurate  to  1',  the  result  may  be 
in  error  by  ^f  ^  days,  or  about  .09  day ;  the  actual  error  is  probably 
less  than  half  this  amount,  and  the  concluded  right  ascensions 
probably  within  10s  of  the  true  values. 

The  observed  times  of  Altair  on  the  two  dates  show  that  the 
clock  was  gaining  46s  daily,  since  the  true  sidereal  time  of  transit, 


88 


LABORATORY   ASTRONOMY 


being  equal  to  the  star's  right  ascension,  is  the  same  on  both  nights. 
This  rate  is  so  large  that  it  cannot  be  neglected  as  in  the  discus- 
sion of  the  result  for  1899. 

If  the  clock  correction  A£  (see  page  60)  at  the  time  of  the  sun's 
transit,  September  22,  be  assumed  0s  and  the  gaining  rate  46s  per 
day,  or  18.916  per  hour,  the  corrections  for  Vega  and  Altair  Sep- 
tember 22  were  — 128.6  and  — 14.9,  and  for  the  sun  and  Altair 
September  23  were  —  45.9  and  —  618.0.  The  times  obtained  by 
applying  these  corrections  are  said  to  be  "  corrected  for  rate  of 
the  clock  to  the  epoch  September  22.0." 

In  this  manner  the  times,  as  they  would  have  been  observed  with 
a  clock  having  an  exact  sidereal  rate,  are  found  to  be  : 


SEPTEMBER  22 

SEPTEMBER  23 

Time  of  transit 

u                    u 
u                    u 

of  the  Sun     .     .     . 
"  Vega     .... 
"   Altair    .... 

Hh    58m  446.8 
18     35      15  .4 
19     47      34.1 

I2h      2m   158.6 
19      47      34  .0 

Hence  Altair  followed  the  sun 

September  22.0  7h  48"  498.3 

23.0  7     45      18.4 

22.856        7     45      48.8 

and  the  right  ascension  of  Altair  was  19h  45m  488.8  ;  since  Vega  pre- 
cedes Altair  by  lh  22m  188.7,  its  right  ascension  was  18h  33m  30s.  1. 

In  1899  the  difference  of  right  ascension  of  the  two  stars  was 
lh  22m  20M,  but  the  right  ascensions  of  1900  are  greater  by  288.3 
and  268.7  than  those  of  1899. 

If  we  assume  the  later  determination  to  be  absolutely  correct, 
we  must  regard  the  earlier  as  having  placed  the  equinox  farther 
toward  the  east  among  the  stars  than  its  true  place,  so  that  right 
ascensions  referred  to  the  equinox  observed  in  1899  are  too  small. 
We  may  say  that  the  observations  of  1900  indicate  a  correction  of 
—  278.5  to  the  "  equinox  of  our  little  catalogue  of  four  stars  " ;  that 
is,  a  correction  of  -f  27s. 5  to  all  their  right  ascensions  as  determined 
in  1899. 


MERIDIAN    OBSERVATIONS  89 

Applying  these  corrections,  their  right  ascensions  become  for 

t]  Serpentis  18h  16ra     7s.  0 

X  Sagittarii  18  21     44  .9 

Vega  18  33     29  .3 

Altair  19  45     49  .4 

Since  the  later  observations  were  made  with  an  instrument 
giving  more  accurate  values  of  the  declination,  it  is  probable  that 
their  results  are  more  nearly  correct.  The  clock  rate  was  neglected 
in  the  first  observations,  and  the  effects  of  precession,  parallax,  and 
refraction  in  both  series,  following  out  the  principle  that  no  correc- 
tions will  be  made  until  observations  shall  show  their  necessity. 

The  effect  of  refraction  is  to  delay  the  autumnal  equinox  about 
an  hour,  and  hence  to  decrease  the  right  ascensions  of  the  stars 
by  about  10s.  At  the  vernal  equinox,  however,  refraction  hastens 
the  equinox  an  hour  and  increases  the  right  ascensions  by  10s; 
its  effect  may  be  shown  by  observations  at  the  two  equinoxes  of 
the  same  year  and  eliminated  by  their  combination.  Parallax 
hastens  the  autumnal  and  delays  the  vernal  equinox  by  about  8m, 
thus  affecting  right  ascensions  by  a  little  more  than  I8,  the  mean 
of  observations  at  the  two  equinoxes  being  free  from  error  from 
this  source.  The  effect  of  precession  will  be  manifest  in  less 
than  ten  years  with  an  instrument  like  that  used  in  the  above 
observations  of  1900. 

By  comparing  the  equinox  of  September  22.714  ±  0.12,  1899,  and 
September  22.856  ±  .09,  1900,  the  length  of  the  tropical  year  is 
found  to  be  365°.142,  but  may  lie  between  364.93  and  365.35  as  far 
as  our  observations  can  surely  determine.  Since  refraction  delays 
the  vernal  and  hastens  the  autumnal  equinox  by  nearly  the  same 
amount  (about  an  hour)  in  each  case,  it  has  no  effect  upon  the 
length  of  the  year.  As  the  greatest  error  to  be  feared  with  our 
improved  instrument  is  less  than  0.1  day,  the  length  of  ten  or  one 
hundred  years  may  be  determined  with  less  than  twice  that  error, 
in  those  periods  the  length  of  the  year  may  be  determined  within 
0.02  and  .002  day,  respectively. 

With  the  best  modern  instrument  used  to  the  greatest  advantage, 
the  sun's  declination  may  be  determined  near  the  equinox  within 


90  LABORATORY    ASTRONOMY 

0".5,  and  hence  the  time  of  the  equinox  within  308  and  right 
ascensions  within  08.08.  A  single  tropical  year  may  be  measured 
with  an  error  of  less  than  lm. 


We  have  now  explained  the  methods  by  which  it  is  possible  to 
fix  the  places  of  the  sun,  moon,  and  stars  at  different  times  and 
thus  to  obtain  data  from  which  their  apparent  motions  about  the 
earth  may  be  studied  and  theories  formed  from  which  their  future 
places  may  be  predicted.  More  or  less  complete  accounts  of  these 
theories  are  to  be  found  in  all  works  on  descriptive  astronomy, 
and  the  predictions  derived  from  them  are  published  for  three 
years  in  advance  by  several  governments  for  the  use  of  navigators 
and  astronomers.  Such  .a  publication  is  the  American  Ephemeris 
and  Nautical  Almanac,  of  which  it  will  be  convenient  to  give  some 
account  before  taking  up  the  motions  of  the  planets. 

The  apparent  motions  of  the  planets  are  less  simple  than  those 
of  the  sun,  moon,  and  stars,  which  at  all  times  seem  to  move  about 
the  earth  as  a  center  with  approximately  uniform  velocities.  The 
planets,  it  is  true,  in  the  long  run  continually  move  like  the  sun 
and  moon  around  the  heavenly  sphere  toward  the  east,  but  their 
velocities  are  variable  within  wide  limits  and  at  certain  times  are 
even  reversed,  so  that  they  move  in  the  opposite  direction  or 
"  retrograde  "  among  the  stars. 

For  this  reason  a  longer  period  of  observation  is  necessary  to 
determine  their  motions  than  can  be  given  by  the  individual  student. 
We  may,  however,  regard  the  nautical  almanacs  of  past  years  as 
predictions  that  have  been  verified,  and  they  stand  for  us  as  an 
accredited  set  of  exceptionally  accurate  observations  from  which 
we  may  draw  material  to  combine  with  the  results  of  our  own 
observations. 


CHAPTEK  VII 
THE  NAUTICAL  ALMANAC 

THE  American  Ephemeris  and  Nautical  Almanac  consists  of  two 
parts, — the  Nautical  Almanac  proper,  which  is  published  separately 
and  contains  data  especially  useful  in  navigation,  and  a  second  part, 
which  contains  additional  tables  adapted  to  the  use  of  astronomers. 
The  Nautical  Almanac  will  suffice  for  most  of  our  purposes,  but  the 
complete  work  is  convenient  for  a  few  references. 

The  tables  contain  data  for  the  sun,  moon,  and  planets,  for  suc- 
cessive equidistant  points  of  Greenwich  mean  time,  so  near  together 
that  the  values  at  any  intermediate  time  may  be  obtained  by  inter- 
polation with  a  degree  of  accuracy  greater  than  can  be  obtained  by 
a  single  observation  made  with  the  most  refined  instruments.  The 
dates  are  given  in  astronomical  time,  each  day  beginning  at  noon 
of  the  corresponding  civil  date. 

At  this  point  a  few  words  are  necessary  in  explanation  of  the  term 
"  mean  time." 

We  have  already  defined  apparent  solar  time  as  the  hour-angle  of 
the  sun,  and  sidereal  time  as  the  hour-angle  of  the  vernal  equinox. 
Owing  to  the  fact  that  the  sun  moves  at  a  varying  angular  rate  and 
in  a  path  inclined  to  the  equinoctial,  the  hour-angle  of  the  sun  does 
not  increase  uniformly,  and  the  hours  of  apparent  time  are,  there- 
fore, of  unequal  length. 

We  have  not  yet  obtained  material  for  a  complete  discussion  of 
the  relation  between  apparent  and  mean  solar  time,  and  for  this  we 
must  refer  to  the  text-books  of  descriptive  astronomy.  It  will 
be  convenient  to  explain  one  simple  statement  of  this  relation  which 
is  not  always  explicitly  given. 

The  time  required  by  the  sun  to  complete  its  circuit  of  the  heavens, 
from  one  passage  through  the  vernal  equinox  to  another,  is  365.2422 
days.  As  it  describes  360°  of  longitude  in  that  time,  its  average 
daily  motion  in  longitude  is  0°. 98564 7. 

91 


92  LABORATORY   ASTRONOMY 

To  establish,  a  convenient  measure  of  time  not  greatly  different 
from  apparent  solar  time,  a  fictitious  body  is  imagined  to  start 
with  the  sun  at  perihelion  and  to  move  along  the  ecliptic  with  a 
uniform  daily  motion  in  longitude  of  0°.9S565.  Its  longitude  at 
any  time  is  called,  appropriately  enough,  the  "  mean  longitude  of 
the  sun." 

When  this  body  reaches  the  vernal  equinox,  a  second  fictitious 
body,  called  the  "  mean  sun,"  is  supposed  to  start  out  from  that 
point  eastward  along  the  equator,  moving  with  a  uniform  velocity 
equal  to  the  mean  daily  motion  of  the  sun  in  the  ecliptic. 

The  mean  sun,  therefore,  continually  increases  its  right  ascension 
by  0°.98565  per  day  ;  and  since  both  fictitious  suns  are  at  the  vernal 
equinox  in  longitude  zero  at  the  same  instant  and  move  at  the  same 
rate,  one  in  the  ecliptic  and  the  other  in  the  equator,  it  is  obvious 
that  at  all  times  the  right  ascension  of  the  mean  sun  is  equal  to 
the  sun's  mean  longitude. 

The  hour-angle  of  the  mean  sun  is  equal  to  the  mean  solar  time, 
just  as  the  hour-angle  of  the  true  sun  is  equal  to  the  apparent  solar 
time. 

A  clock,  properly  regulated  and  set  so  that  it  shows  Oh  Om  0s 
at  each  successive  passage  of  the  mean  sun  over  the  meridian  of 
a  given  place,  is  said  to  keep  the  local  mean  time  of  that  place. 
When  the  hour-angle  of  the  mean  sun  is  10°,  20°,  30°,  the  local 
mean  time  is  Oh  40m,  lh  20m,  2h,  respectively. 

It  is  of  course  true  of  the  mean  sun  as  of  any  other  heavenly 
body  (see  page  58)  that  its  H.A.  +  R.A.  =  Sid.  T.  We  may  there- 
fore write : 

H.A.  of  mean  sun  -f  K.A.  of  mean  sun  =  Sid.  T. 
H.A.  of  sun  -f  K  A.  of  sun  =  Sid.  T. 

And  from  these  equations,  remembering  the  definitions  of  mean 
and  apparent  time,  we  derive  the  following : 

Mean  T.  =  App.  T.  +  (E.A.  of  sun  —  E.A.  of  mean  sun). 

The  quantity  in  the  parenthesis,  which  must  be  added  to  App.  T. 
to  give  the  corresponding  Mean  T.,  is  called  the  equation  of  time. 


THE   NAUTICAL   ALMANAC  93 

The  equation  of  time  is  the  difference  between  mean  time  and 
apparent  time,  and  when  positive  must  be  added  to  apparent  time 
to  give  the  corresponding  mean  time,  or  subtracted  from  mean  time 
to  find  the  corresponding  apparent  time. 

Standard  Time.  —  It  is  now  usual  to  regulate  the  clocks  over  large 
sections  of  country  to  the  mean  time  of  a  neighboring  meridian. 
Thus,  clocks  in  the  central  part  of  the  United  States  are  set  to 
show  Oh  Om  0s  when  the  sun  is  in  the  meridian  whose  longitude 
is  90°  west  of  Greenwich,  and  they  are  said  to  keep  Central 
standard  time ;  which  is,  therefore,  6  hours  slow  of  Greenwich 
time.  Other  sections  use  the  mean  time  of  the  75th,  105th,  and 
120th  meridians,  5,  7,  and  8  hours  slow  of  Greenwich,  respectively. 
More  than  one  half  the  people  of  the  United  States  use  Central 
standard  time. 

The  fact  that  our  watches  are  set  to  standard  time  is  a  convenience 
in  using  the  Almanac,  since  the  watch  time  gives  us  Greenwich  mean 
time  by  applying  so  simple  a  correction,  the  minutes  and  seconds 
being  unchanged  and  the  hours  increased  by  a  small  constant  number. 


THE  CALENDAR 

About  four-fifths  of  the  Nautical  Almanac  consists  of  data  regard- 
ing the  sun  and  moon,  eighteen  successive  pages  being  devoted  to 
each  month,  and  the  corresponding  pages  of  the  different  months 
numbered  with  the  Roman  numerals  from  I  to  XVIII.  These  pages, 
which  form  the  Calendar,  we  will  now  consider  in  detail.  The 
reading  matter  of  the  Explanation  which  follows  the  tables  should 
be  carefully  read  in  connection  with  the  following  paragraphs : 
reduced  facsimiles  of  several  pages  are  shown  at  page  176,  to  which 
reference  may  be  made. 

The  positions  are  given  as  they  would  appear  to  an  observer  at 
the  earth's  center,  and  the  times  are,  as  stated  at  the  head  of  each 
page,  Greenwich  mean  time.  We  pass  at  once  to  page  II,  which, 
rather  than  the  very  similar  page  I,  it  will  be  always  more  con- 
venient to  use  when,  as  in  most  of  our  observations,  the  Greenwich 
time  is  known  for  which  the  data  are  required. 


94  LABORATORY   ASTRONOMY 

Page  II. — The  first  and  second  columns  give  the  day  of  the  week 
and  month.  The  third  column  contains  the  sun's  apparent  right 
ascension  at  Gr.  Mean  Noon,  —  that  is,  its  right  ascension  as  affected 
by  the  annual  aberration  (which  makes  it  appear  to  be  about  20" 
behind  its  true  place  in  its  orbit)  and  measured  from  the  actual 
equinox  of  the  date.  Column  4  contains  the  hourly  difference,  or 
the  amount  by  which  the  right  ascension  is  changing  per  hour. 

To  illustrate  the  use  of  this  column,  let  it  be  required  to  find 
the  right  ascension  of  the  sun  at  the  time  of  the  first  observa- 
tion recorded  on  page  39  at  8h  54m  37s  A.M.,  Eastern  standard  time, 
March  8,  1900. 

We  must  first  notice  that  the  corresponding  astronomical  time, 
which  is  reckoned  from  noon  to  noon,  is  20h  54m  37s  after  noon  of 
the  preceding  day,  —  that  is,  the  local  date  was  March  7d  20h  54m  37s ; 
adding  5h  to  change  E.  Std.  T.  to  G.M.T.,  we  have  March  7d25h 
54ra  37s,  or  March  8d  lh  54m  37s,  G.M.T. 

The  sun's  right  ascension,  March  8,  at  Greenwich  mean  noon,  is 
given  as  23h  13m  578.68.  To  this,  since  the  sun's  right  ascension  is 
always  increasing,  must  be  added  the  change  in  lh  54m  37s  (=  lh.91), 
the  time  elapsed  since  noon,  which  is  obtained  by  multiplying  the 
hourly  difference  found  in  column  4  by  1.91 ;  this  gives  the  correc- 
tion to  be  added  to  the  tabular  right  ascension  as  1.91  x  9S.237, 
or  178.64,  and  the  right  ascension  at  the  time  of  observation  was 
therefore  23h  13m  578.68  +  178.64,  or  23h  14ra  15S.32. 

This  simple  process,  which  is  fully  illustrated  in  the  Explanation, 
will  never  give  a  value  more  than  08.4  in  error.  A  method  of  inter- 
polation by  which  an  accuracy  of  08.01  may  be  attained  is  given  in 
the  Explanation.  The  error  of  the  simple  method  arises  from  the 
fact  that  the  hourly  difference  is  not  constant,  as  will  appear  at 
once  from  inspection  of  the  values  in  the  fourth  column. 

Columns  5  and  6  give  the  sun's  apparent  declination  and  its 
hourly  difference.  The  value  at  any  time  may  be  found  by  inter- 
polation in  the  manner  just  explained. 

North  declinations  are  regarded  as  positive,  and  south  decli- 
nations negative,  and  in  accordance  with  this  convention  the  hourly 
difference  is  marked  +  when  the  change  of  declination  is  toward 
the  north  and  —  when  toward  the  south,  so  that  the  true  declination 


THE   NAUTICAL  ALMANAC 


95 


is  found  by  applying  the  correction  algebraically :  thus,  to  find  the 
declinations  at  4  P.M.,  G.M.T.,  on  the  following  dates,  we  have : 


1900 

8  AT  MEAN  NOON 

H.  DlFF. 

CORK.  FOR  4h 

S  AT  4»>  G.M.T. 

Jan.    10 

-21°  59'   4".0 

+  22".  25 

+  4  x  22".  25  =  +    89".0 

-  21°  57'  35".0 

April  10 

+    7  53    3  .7 

+  55  .48 

+  4  x  55  .48  =  +  221  .9 

+    7    56  45  .6 

Aug.  10 

+  15  38  18  .2 

-43  .73 

-4  x43  .73  =  -174  .9 

+  15   35  23  .3 

Nov.    10 

-  17     6  18  .2 

-42  .31 

-4  x42  .31  =-169  .2 

-  17     9.7  .4 

The  error  in  a  declination  determined  by  a  simple  interpolation 
from  the  preceding  mean  noon  can  never  exceed  12".  By  the  more 
accurate  method  given  in  the  Explanation,  it  is  always  less  than  0".l. 

To'  make  sure  that  the  correction  has  been  applied  with  the  proper 
sign,  it  is  sufficient  to  notice  that  the  computed  value  must  lie 
between  the  values  for  the  including  dates. 

Columns  7  and  8  contain  the  equation  of  time  and  its  hourly 
difference.  The  correction  to  be  applied  is  obtained,  as  in  the  pre- 
ceding examples,  by  multiplying  the  hourly  difference  by  the  num- 
ber of  hours  elapsed  since  Greenwich  mean  noon,  and  must  either 
be  added  or  subtracted  so  as  to  give  a  value  between  the  values  of 
the  including  dates. 

The  heading  of  the  column  indicates  whether  the  equation  of  time 
is  to  be  added  to  or  subtracted  from  mean  time  to  give  apparent 
time.  Of  course  when  it  is  additive  to  mean  time  it  must  be  sub- 
tracted from  apparent  time  to  give  mean  time,  as  will  appear  on 
comparing  the  corresponding  column  of  page  I. 

Example.  What  is  the  equation  of  time  January  10,  1900,  at 
3h  45m,  Central  standard  time  ? 

The  corresponding  G.M.T.  is  9h  45m  =  9h.75 

Eq.  of  T.  at  Gr.  Mean  Noon      .     +  7m  39s.  87  H.  Diff.  =  K014 

Change  in  9h.75  =  9.75  x  18.014                9  .88  x  9.75 

Eq.  of  T.  at  3h 45m,  Cent.  T.     .     +7    49.75  Corr.  =98.88 

The  correction  9s.  88  is  added  because  the  value  of  the  equation 
January  11  is  seen  to  be  8m  38.90,  and  the  correction  must  be  applied 
so  as  to  increase  numerically  the  value  on  January  10. 


96  LABORATORY   ASTRONOMY 

The  ninth  column  contains  the  right  ascension  of  the  mean  sun. 
Since  at  mean  noon  the  mean  sun  is  on  the  meridian  and  since 
(p.  59)  the  right  ascension  of  a  body  which  is  on  the  meridian  at 
a  given  instant  equals  the  sidereal  time  at  that  instant,  the  right 
ascension  of  the  mean  sun  at  Greenwich  mean  noon  equals  the 
Greenwich  sidereal  time  at  Greenwich  mean  noon,  and  this  explains 
the  alternative  heading  which  appears  at  the  top  of  the  column. 

Since  the  right  ascension  of  the  mean  sun  increases  uniformly, 
the  constant  hourly  difference  requires  no  special  column,  but  is 
given  at  the  foot  of  the  page.  For  interpolation  it  is  most  con- 
venient to  use  Table  III,  which  occupies  three  of  the  last  pages  of 
the  Almanac,  and  gives  directly  the  multiples  of  98.8565  by  each  hour 
and  minute  up  to  24  hours,  thus  saving  the  reduction  of  minutes  to 
decimals  of  an  hour. 

Example.  Right  ascension  of  mean  sun,  January  15,  1900,  at 
4h  44m  30s. 

R.A.  mean  sun,  Gr.  Mean  Noon   .     ...     .     .     .     .     .     19h  37m  55s.  26 

Add  4M4m  30s  x  9s.  8505  (Table  III)       .     .     .     .     .  46.74 

E.A.  meansunat4h44m308 19   38    42  .00 

This  is  obviously  the  sidereal  time  of  mean  noon  at  a  place  in 
longitude  4h  44m  308  west,  and  if  desired  a  table  of  this  quantity 
may  be  computed  for  such  a  place  by  adding  468.74  to  the  values 
given  each  day  in  the  Almanac  for  Greenwich. 

Page  I.  —  The  quantities  on  page  I  are  only  used  for  reducing 
meridian  observations  of  the  sun,  which  are  made,  of  course,  at  local 
apparent  noon.  This  page  is  convenient  when  the  Greenwich  mean 
time  has  not  been  noted,  for  the  time  elapsed  since  the  preceding 
Greenwich  apparent  noon  is  equal  to  the  west  longitude  of  the 
place  of  observation.  This  is  the  quantity,  therefore,  by  which  the 
hourly  difference  must  be  multiplied  to  give  the  correction.  An 
example  of  the  use  of  this  page  is  given  on  page  104. 

All  the  quantities  given  on  page  I  may  be  found  more  easily  from 
page  II  if  we  know  the  G.M.T.  for  which  they  are  required.  The 
only  quantity  for  which  we  are  obliged  to  consult  page  I  is  the 
semi-diameter,  and  this  never  differs  by  so  much  as  0".01  from  its 
value  at  mean  noon. 


THE   NAUTICAL   ALMANAC  97 

Page  III.  —  Column  2  gives  the  day  of  the  year  corresponding  to 
the  given  date,  and  is  convenient  for  finding  the  number  of  days 
intervening  between  dates.  Thus,  January  15,  1900,  is  the  15th 
day  of  the  year  and  September  25  is  the  268th ;  hence  from  noon, 
January  15,  to  noon,  September  25,  is  268  — 15,  or  253  days. 

Column  3  contains  the  sun's  longitude  measured  from  the  vernal 
equinox  of  the  given  date.  For  some  purposes  it  is  more  convenient 
to  measure  from  the  mean  equinox  of  the  beginning  of  the  fictitious 
year,  an  epoch  much  used  in  astronomical  calculations  but  of  no 
intrinsic  interest.  The  minutes  and  seconds  of  the  longitude  as 
thus  measured  are  found  in  column  4.  The  longitude  of  column  3 
is  measured  from  the  actual  place  of  the  equinox  at  the  given  date 
as  affected  by  precession  and  nutation. 

Column  6  gives  the  sun's  latitude,  which  is  always  nearly  but 
not  exactly  zero,  as  will  be  explained  further  on  in  this  chapter. 

Column  7  gives  the  logarithm  of  the  earth's  distance  from  the 
sun  in  astronomical  units.  An  astronomical  unit  is  equal  to  the 
semi-axis  major  of  the  earth's  orbit,  —  about  93,000,000  miles. 
For  those  unacquainted  with  logarithms  the  following  table  will 
make  it  easy  to  find  by  interpolation  the  approximate  distance  cor- 
responding to  a  given  logarithm. 

Logarithm  9.9925000  corresponds  to  0.9829  astronomical  units. 

"  9.9950000  "  "  0.9886  "  " 

"  9.9975000  "  "  0.9943  "  " 

"  0.0000000  "  «'  1.0000  "  " 

"  0.0025000  "  "  1.0058  "  " 

"  0.0050000  "  "  1.0116  "  " 

"  0.0075000  "  "  1.0174 

Example.  January  19,  1900,  log  radius  vector  =  9.99299,  which 
is  very  nearly  ^  of  the  way  from  9.9925  to  9.9950 ;  hence  on  that 
date  the  distance  of  the  earth  from  the  sun  is  £  of  the  way  between 
0.9829  and  0.9886,  or  0.9840  astronomical  units.  The  value  can  be 
obtained  within  less  than  ^1^  of  its  amount  without  interpolation 
by  taking  the  nearest  value  of  the  logarithm  given  in  the  table. 

Column  9  gives  the  mean  time  at  which  the  vernal  equinox  is  on 
the  meridian  of  Greenwich  (when  the  number  of  hours  is  greater 
than  12  the  time  is  after  midnight,  and  therefore  during  the  morning 


98  LABORATORY   ASTRONOMY 

hours  of  the  next  civil  date).  This  quantity  is  sometimes  used  in 
converting  sidereal  to  mean  time,  but  its  use  may  be  easily  avoided 
and  is  sufficiently  treated  in  the  Explanation. 

Page  IV.  —  The  quantities  on  page  IV  relate  to  the  moon.  They 
are  given  for  each  12  hours  of  Greenwich  mean  time,  and  seem  to 
call  for  no  explanation,  except  perhaps  the  symbol  6,  signifying 
conjunction,  which  occurs  once  (and  occasionally  twice)  upon  each 
page,  on  the  day  before  or  after  that  of  new  moon.  Since  successive 
transits  follow  each  other  nearly  25  hours  apart,  in  general  one  date 
in  each  month  would  be  left  blank,  the  moon  crossing  the  meridian 
during  the  hour  preceding  noon  of  one  date,  and  during  the  hour 
following  noon  of  the  succeeding  date.  The  symbol  6  occupies  the 
vacant  space  and  marks  the  date  of  new  moon. 

Pages  V  to  XII  contain  the  right  ascension  and  declination  of  the 
moon  for  every  hour  of  G.M.T.,  together  with  their  differences  for 
each  minute  of  time.  The  rapid  motion  of  the  moon  makes  it  necessary 
to  give  these  quantities  at  shorter  intervals  than  suffice  for  the  sun, 
in  order  that  an  equal  accuracy  may  be  attained  in  interpolation. 

These  are  of  course  places  as  seen  from  the  earth's  center,  and 
it  is  to  be  remembered  that  at  any  point  on  the  earth's  surface  the 
moon  may  be  displaced  by  parallax  a  little  more  than  1°. 

On  page  XII  are  given  the  exact  dates  to  the  nearest  hour  of 
G.M.T.  of  the  moon's  phases  and  the  times  of  perigee  and  apogee. 

Pages  XIII  to  XVIII  contain  tables  of  "  lunar  distances,"  —  that 
is,  distances  for  each  three  hours  of  Greenwich  mean  time  between 
the  moon's  center  and  certain  bright  stars  and  planets  not  far  from 
the  plane  of  its  motion ;  the  sun  is  included  in  the  list,  as  the  moon 
is  often  visible  in  full  daylight,  so  that  its  distance  from  the  sun 
may  be  easily  measured. 

This  table  is  used  in  determining  longitude  ;  the  local  time  being 
known,  the  G.M.T.  may  be  found  by  the  method  of  lunar  distances, 
as  follows :  The  distance  from  moon  to  star  or  sun  being  measured 
is  found  to  lie  between  two  distances  given  in  the  table  ;  the  G.M.T. 
of  the  observation  then  lies  between  the  hours  corresponding  to  the 
two  tabular  distances,  and  its  exact  value  may  be  determined  by 
interpolation.  The  difference  between  this  time  and  the  known 
local  time  of  the  observation  is  the  longitude. 


THE   NAUTICAL   ALMANAC  99 

The  method  requires  accurate  observations,  and  troublesome  com- 
putations are  necessary  to  correct  the  measured  distance  for  the 
effects  of  refraction  and  parallax  so  as  to  find  the  distance  from 
moon  to  star  as  seen  from  the  earth's  center. 

Data  for  the  Planets,  Eclipses.  —  Following  the  calendar  pages  of 
the  Nautical  Almanac  are  thirty  pages  giving  the  right  ascension 
and  declination  and  the  time  of  meridian  passage  of  the  five  planets 
which  are  visible  to  the  naked  eye,  and  three  pages  containing  the 
right  ascensions  and  declinations  of  150  of  the  brighter  fixed  stars. 

A  few  pages  are  devoted  to  the  eclipses  of  the  year,  with  maps 
from  which  may  be  obtained  the  approximate  times  of  the  successive 
phases  of  the  solar  eclipses  as  seen  from  any  given  point  of  obser- 
vation on  the  earth. 

EXAMINATION  OF   THE   SEVERAL   COLUMNS 

Having  given  this  general  summary  of  the  contents  of  the  tables, 
we  will  now  call  attention  to  some  of  the  interesting  facts  and  rela- 
tions that  appear  on  running  through  the  various  columns  throughout 
the  whole  year. 

The  date  of  the  solstices  may  be  determined  as  the  days  on  which 
the  sun's  declination  has  its  maximum  northern  and  southern  values. 

The  date  of  the  equinoxes  may  be  found,  from  either  the  right 
ascension  or  declination  columns,  as  the  date  on  which  the  decli- 
nation changes  sign,  and  the  right  ascension  is  either  Oh  or  12h;  the 
exact  time  may  be  found  by  interpolation.  (See  page  107.) 

The  number  of  days  between  the  equinoxes  may  be  determined 
by  using  the  column  of  days,  page  III.  It  will  be  found  that  the 
sun  is  for  some  days  more  than  half  the  year  in  that  part  of  its 
orbit  which  lies  in  the  northern  hemisphere. 

The  column  of  hourly  difference  shows  that  the  declination  is 
changing  slowly  at  the  solstices  and  most  rapidly  at  the  equinoxes ; 
moreover,  the  change  at  the  latter  dates  is  nearly  uniform  both  in 
right  ascension  and  declination,  as  stated  on  page  85.  If  a  right 
triangle 'be  drawn  with  the  difference  in  right  ascension  for  the 
date  of  the  equinox  as  base  and  difference  in  declination  as  alti- 
tude, the  angle  between  the  base  and  the  hypotenuse  measured  by 


100  LABORATORY  ASTRONOMY 

a  protractor  will  be  found  to  be  23^°.  It  obviously  equals  the  angle 
between  the  equator  and  the  ecliptic. 

Notice  that  the  equation  of  time  is  the  difference  between  right 
ascension  of  mean  and  true  sun,  as  stated  on  page  92,  thus  : 

From  the  Almanac  for  1900  (p.  II),  we  have  the  following 
values  :  January  21,  Sun's  B,. A.  =  20h  13m  28.79 ;  K.A.  Mean  Sun 
—  20h  lm  34s. 61.  Subtracting  the  latter  from  the  former,  we  have  for 
the  equation  of  time  +  llm  288.18.  This  is  the  value  given  on  page  II ; 
the  positive  sign  indicates  that  it  is  to  be  added  to  apparent  time  to 
find  mean  time,  or  subtracted  from  mean  time  to  find  apparent  time. 

The  dates  on  which  the  equation  of  time  is  0  and  dates  and  values 
of  greatest  and  least  equations  should  be  noticed ;  also  that  on  those 
dates  for  which  the  equation  is  0  the  values  of  the  sun's  right  ascen- 
sion and  declination,  etc.,  on  pages  I  and  II,  are  the  same,  since 
apparent  noon  and  mean  noon  coincide.  For  1900  the  civil  dates 

are  as  follows  : 

EQ.  OF  T. 

February  11 +  14m  278.28 

April  15 0 

May  15 -    3m  498.40 

June  14 0 

July  27 r    .     .     .     .     ...     .".  +    6m  178.22 

September  1 0 

November  3 -  16m  20s. 40 

December  25 0 

The  hourly  difference  of  the  right  ascension  of  the  mean  sun  has 
the  same  integers  as  the  mean  daily  motion  of 'the  sun  in  longitude, 

0°.98565         0.98565  X  3600" 
0.98565  ;    for  0  .98565  per  day  =  — — ,  or  -  — ,  per 

hour,  and  reducing  this  to  seconds  of  time  by  dividing  by  15,  we 
find  the  motion  of  the  mean  sun  to  be  98.8565  per  hour.  This  illus- 
trates the  fact  that  the  mean  motion  of  the  sun  in  longitude  (0°. 98565 
per  day)  is  the  same  as  that  of  the  mean  sun  in  right  ascension 
(98.8565  per  hour),  page  92. 

The  column  which  gives  the  sun's  latitude  will  repay  an  investi- 
gation. It  appears  at  a  glance  that  there  is  a  small  but  regular 
change,,  from  south  to  north  and  return,  with  a  period  of  about  27 
or  28  days. 


THE   NAUTICAL   ALMANAC  101 

The  principal  cause  of  this  is  that  it  is  not  the  earth,  but  the 
center  of  gravity  of  the  earth  and  moon,  which  describes  an  orbit 
in  the  plane  of  the  ecliptic ;  and  by  the  known  properties  of  the 
center  of  gravity,  when  the  moon  is  above  the  ecliptic  the  earth 
must  be  below.  It  is  not  very  difficult  to  show  that  from  this  cause 
the  latitude  may  be  0".67  greater  or  less  than  when  both  bodies  are 
in  the  ecliptic,  that  is,  when  the  moon  is  at  one  of  her  nodes. 

The  attractions  of  Venus  and  Jupiter  also  draw  the  earth  out  of  the 
ecliptic  by  an  amount  which  may  reach  0".5.  In  January,  1900,  this 
"  planetary  perturbation  "  was  about  +  0".13.  The  total  range  of  lat- 
itude during  the  month  (see  page  178)  was  from  -f  0".68  to  —  0".48. 
The  moon  was  at  her  nodes  January  12.33  and  January  26.85. 

From  the  radius  vector  column  (p.  Ill)  we  may  find  the  sun's 
distance  at  any  date  by  the  table  on  page  97.  By  comparing  this 
with  the  semi-diameter  column  (p.  I),  it  is  shown  that  the  sun's 
distance  is  inversely  proportional  to  its  angular  semi-diameter. 
Thus,  January  1,  1904  : 

Log  r  =  9.9926540,  Dist.  =  0.9832,  Senii-diam.  =  16'  17".90 
and  July  1,  1904  : 

Log  r  =  0.0072095,  Dist.  =  1.0167,  Semi-diam.  =  15'45".67 
and 

0.9832  : 1.0167  =  945".67 :  977".90, 

as  appears  on  multiplying  the  means  and  extremes  and  comparing 
the  products. 

The  dates  of  the  moon's  perigee  and  apogee  may  be  determined 
from  the  greatest  and  least  semi-diameter,  page  IV,  column  2,  or 
from  the  greatest  and  least  parallax  in  column  4.  Since  both  semi- 
diameter  and  parallax  are  inversely  proportional  to  the  moon's 
distance  from  the  earth,  the  latter  may  be  determined  by  multiplying 
the  former  by  a  constant  quantity.  This  constant  is  3.6625,  and  it 
is  not  difficult  to  show  that  it  is  the  ratio  of  the  earth's  equatorial 
radius  to  that  of  the  moon. 

Compare  the  last  two  columns,  noting  that  at  new  rnoon  the  moon 
comes  to  the  meridian  with  the  sun  at  noon  and  that  at  full  moon 
(age  15  days)  it  comes  to  the  meridian  at  midnight. 


102  LABORATORY  ASTRONOMY 

TABLES  OF  THE  PLANETS  AND  STARS 

The  data  for  the  planets  which  follow  the  calendar  pages  illus- 
trate many  facts  which  are  explained  in  the  text-books  on  descriptive 
astronomy. 

Ketrograde  motion,  for  example,  is  shown  by  negative  hourly 
differences  in  right  ascension ;  the  stationary  points  occur  on  those 
dates  on  which  the  hourly  difference  changes  sign  ;  opposition  takes 
place  when  the  time  of  transit  is  12h;  conjunction,  when  it  is  Oh ; 
the  retrograde  motion  is  a  maximum  at  opposition. 

By  means  of  the  right  ascensions  and  declinations  the  path  for 
the  year  may  be  plotted  on  a  star  map,  for  which  purpose  an  ecliptic 
map  (see  page  65)  is  especially  adapted. 

The  time  of  passing  the  node  may  be  found  from  the  point  where 
the  path  cuts  the  ecliptic,  and  the  sidereal  period  from  the  interval 
between  two  passages  of  the  same  node. 

A  series  of  Almanacs  covering  some  years  is  useful  in  following 
the  outer  planets  as  well  as  for  comparison  of  the  calendar  pages  to 
show  the  repetition  of  the  solar  data  after  four  years. 

The  table  of  star  places  contains  columns  of  annual  variation,  — 
that  is,  the  sum  of  the  precession  and  proper  motion  (the  latter 
always  a  very  small  quantity),  —  which  are  useful  in  showing  the 
effects  of  precession  on  the  right  ascensions  and  declinations  of 
stars-  in  different  parts  of  the  heavens.  Compare  in  this  respect 
8  Draconis,  (3  Ursae  Minoris,  Polaris,  y  Pegasi,  y  Geminorum,  and 
A.  Sagitarii. 


COMPARISONS    OF    OBSERVATIONS    WITH    THE    EPHEMERIS 

Many  of  the  facts  which  we  have  obtained  by  observation  in 
former  chapters  may  be  found  in  the  columns  of  the  Almanac,  and 
after  a  thorough  comprehension  of  the  methods  has  been  acquired 
much  time  may  be  saved  by  employing  these  data ;  but  it  is  to  be 
remembered  that  facts  thus  obtained  are  not  so  thoroughly  grasped 
or  so  easily  retained.  With  this  caution,  we  may  compare  some  of 
the  results  of  our  previous  work  with  the  tables,  to  give  an  idea  of 
the  methods  of  using  the  latter.  Following  are  comparisons  of  a 


THE  NAUTICAL  ALMANAC  103 

few  of  the  observations  of  the  preceding  chapters  with  the  values 
given  by  the  Ephemeris  : 

Observations  of  the  Moon.  —  From  careful  measurement  of  the  map 
on  page  29,  the  moon's  declination  on  January  9,  1900,  at  10 
P.M.,  was  +  19°.3,  and  its  right  ascension  was  2h  38m.  The  place  of 
observation  was  4h  44m.5  west  of  Greenwich,  and  the  time  used  was 
Eastern  standard  time,  which  is  5  hours  slow  of  Greenwich ;  the 
G.M.T.  was  therefore  15h  Om,  at  which  time. the  moon's  declination 
and  right  ascension  are  given  in  the  Ephemeris  (p.  180)  as  +  18°  48' 
and  2h  39m.  The  difference  between  the  observed  and  calculated 
places  is  about  ^°  -in  declination  and  lm  in  right  ascension,  mainly 
due  to  error  of  observation  with  the  cross-staff. 

Length  of  the  Month.  —  We  may  use  the  Ephemeris  to  find  the 
length  of  the  month  by  seeking  the  next  date  at  which  the  moon's 
right  ascension  and  declination  are  the  same,  which  is  February  5, 
at  about  21  hours,  G.M.T.,  as  will  be  seen  from  page  VI  for  February. 
This  gives  27d  6h  as  the  period  of  the  moon's  revolution  among  the 
stars. 

Passing  to  page  V  for  December,  we  find  that  the  right  ascension 
was  again  2h  39m  on  December  3  at  19  hours,  at  which  time  the 
declination  was  17°  19'.  This  shows  that  the  moon's  orbit  had 
shifted  during  this  time  so  that  it  did  not  pass  through  exactly  the 
same  points  of  the  heavens  in  these  two  months,  its  December  path 
in  the  neighborhood  of  right  ascension  2h  39m  being  l£°  south  of 
the  corresponding  point  of  its  path  in  January. 

By  column  2  of  page  III,  January  9  is  the  9th  day  of  the  year 
and  December  3  is  the  337th  ;  hence  the  moon  completed  an  integral 
number  of  revolutions  in  337d  19h  -  9d  15h,  or  328d  4h. 

The -period  having  been  determined  as  27£  days  approximately 
and  328  -s-  271  being  nearly  12,  it  is  evident  that  the  number  of 
complete  revolutions  between  these  dates  is  12.  Dividing  328d  4h 
by  12,  we  have  27d  8h  as  a  closer  approximation  to  the  sidereal 
month. 

Taking  the  length  of  the  successive  months  during  the  year,  it  is 
interesting  to  note  how  very  considerable  is  the  difference  in  length 
of  the  successive  sidereal  months  due  to  the  "  perturbations "  of 
the  moon's  motion. 


104  LABORATORY  ASTRONOMY 

Observations  at  Apparent  Noon.  —  The  observations  recorded  on 
page  39  were  made  at  Cambridge,  in  longitude  4h44m.5  west  of 
Greenwich,  and  the  watch  time  of  apparent  noon  was  llh  56m  28.9. 

By  the  use  of  the  Almanac,  we  find  the  correction  of  the  watch 
to  standard  time  as  follows  : 

Since  the  observation  was  made  at  local  apparent  noon,  it  will 
be  better  to  use  page  I  of  the  Almanac,  which  gives  for  March  8, 
at  Greenwich  apparent  noon,  equation  of  time  llm  18.46,  to  be  added 
to  apparent  time,  and  hourly  difference  0s.  61 9. 

The  time  of  observation  was  4h  44m.5,  or  nearly  4h.75  later,  and 
the  change  of  the  equation  of  time  in  this  interval  was  4.75  X  0s. 619 
=  28.93.  As  the  equation  of  time  was  decreasing,  its  value  at  the 
time  of  observation  was  10m  5SS.53.  Since  no  sign  is  appended  to 
the  hourly  difference,  we  check  this  result  by  noting  that  it  falls 
between  the  values  tabulated  for  March  8  and  9.  Hence : 

Camb.  App.  T.     .     .     . 12h  Om    0s 

Eq.  of  T.  (add) 10     58  .53 

Camb.  M.T. 12  10     58.53 

Corr.  for  Long 4  44     30 

G.  M.T.  of  observation 16  55     28.53 

Subtracting 500 

Eastern  Std.  T.  of  observation 11  55     28.53 

Observed  watch  time     .     .     .     .     ...     .     ...  11  56       2.9 

Corr.  of  watch  to  Std.  T.    .     .     .     .     .   ....     .     .     .     .  -  34 .37 

The  correction  for  longitude  to  give  G.M.T.  is  added,  because  at 
any  given  instant  the  local  time  of  any  place  is  greater  than  that 
of  a  place  to  the  westward,  since  the  sun  passes  its  meridian  earlier 
and  always  has  a  greater  hour-angle  than  at  the  western  place. 

Kemembering  that  Cambridge  is  15m  30s  east  of  the  meridian 
from  which  Eastern  standard  time  is  reckoned,  we  may  find  the 
watch  correction  more  simply,  thus  : 

Camb.  M.T 12h     10m  58S.53 

Reduction  for  Long,  (subtract) -  15     30 

Eastern  Std.  T 11       55     28  .53 

Watch  time 11       56       2  .9 

At       -  34  .37 

Observations  of  the  Planets.  —  The  data  on  page  52  show  that  on 
February  5,  1900,  at  7h  12m  (the  watch  keeping  Eastern  standard 


THE  NAUTICAL  ALMANAC 


105 


time),  the  right  ascension  of  Venus  was  9°.64  =  38m  338.6  less  than 
that  of  y  Pegasi,  which  from  the  Ephemeris  was  Oh  8m  5S.69 ;  hence 
from  this  differential  observation  the  right  ascension  of  Venus  was 
23h29m32s.09. 

The  G.M.T.  of  the  observation  was  12h  12m  =  12h.2. 

The  tables  for  Venus  (p.  224)  give  : 


H.  DlFFS. 

+  11s.  106  +77".  31 

x  12.2  x  12  .2 

135.49  943  .2 

2m158.49  15'  43".  2 


The  observation  differs  from  the  Ephemeris  by  lm  38s  in  right 
ascension  and  8'  in  declination,  although  the  method  should  give 
angles  within  0°.2.  The  discrepancy  is  much  greater  than  usually 
occurs,  and  this  observation  of  Venus  is  affected  by  some  unexplained 
error  ;  it  depends  on  a  single  reading  of  the  hour-angle.  To  exhibit 
the  usual  accuracy,  we  may  compare  with  the  following  observa- 
tions, made  February  6 : 


FEBRUARY  5  R.A.  OF  VENUS 
At  Gr.  M.  noon  .  .  23h25m388.14 
Diff.  for  12h.2  .  .  +2  15.49 

DECL. 
-4°  55'  46" 
+  15  43 

At  12M2m,  G.M.T.  .  2327  53.63 
Observed  values  (p.  53)  23  29  32  .1 

-4  40     3 
-4  32 

WATCH  TIME 

H.A. 

DECL. 

7  Pegasi      .     .     .     .     . 
Venus     

7h    3m    IQs 

5     10 

64°.  6 
74  .05 

+  15°.  45 
-    3.4 

7  Pegasi      ...     .     .     . 

7     10 

65  .7 

+  15  .5 

Hence  Venus  preceded  y  Pegasi  8°.90  =  35m  36s,  Decl.  =  —  3°.4 
—  0°.53  =  —  3°.93 ;  and  since  the  right  ascension  of  y  Pegasi  was 
Oh  8m  6s,  our  observation  gives  for  the  place  of  Venus  at  12h  5m 
G.M.T.,  E.  A.  =  23h  32m  30s,  and  8  =  -  3°  56'.  The  Ephemeris  gives 
E.A.  =  23h  32m  17S.2,  and  8  =  —  4°  9'  14".5. 

Observations  of  the  Moon's  Place. — The  data  given  on  page  55 
show  that  on  February  6,  1900,  the  moon  followed  y  Pegasi  46°.7 
=  3h  6m  48s.  The  right  ascension  of  y  Pegasi  was  Oh  8m  6s ;  hence 
the  moon's  right  ascension  was  3h  14m  54s,  while  its  declination, 
given  directly  by  the  circle,  was  +  20°. 4.  The  Eastern  standard 
time  was  7h  42m,  corresponding  to  12h  42m  G.M.T. 


106  LABORATORY  ASTRONOMY 

The  Ephemeris  gives  : 

MOON'S  K.A.  DECL.  DIFFS.  FOB  lm 

At  12h  G.M.T 3h14m348  +20°  25'  +    2s. 33         +    6". 2 

Diff.  for  42m +  1    38  +  4  sx  42              x  42 

At  time  of  observation      .     .     3    16    12  +  20   29  97  .9              260.4 

Observed  values 3    14    54  +  20  24  Im378.9          +  4' 20" 

The  agreement  here  is  satisfactory  considering  that  the  moon  is 
more  than  45°  from  the  star  with  which  it  is  compared.  Part  of 
the  difference  is  due  to  parallax. 

Observations  of  the  Sun's  Place.  —  By  the  observation  treated  on 
page  67,  the  sun's  right  ascension  and  declination  at  5h  36m  26s, 
Cambridge  sidereal  time,  March  29, 1899,  by  comparison  with  a  Ceti, 
were  found  to  be  Oh  33m  19s  and  +  3°.6.  To  compare  this  with  the 
Ephemeris  of  the  sun,  we  must  first  find  the  Greenwich  mean  time 
corresponding  to  5h  36m  268,  Cambridge  sidereal  time.  Heretofore 
we  have  had  given  either  local  apparent  time  or  standard  time  of 
observations,  and  the  Greenwich  mean  time  has  been  found  by  adding 
the  equation  of  time  and  longitude  in  one  case  or  an  integral  number 
of  hours  in  the  other.  In  this  case  we  have  given  the  local  sidereal 
time,  to  find  the  corresponding  Greenwich  mean  time. 

The  first  step  is  to  find  the  Greenwich  sidereal  time  by  adding  the 
longitude  west  of  Greenwich,  after  which  G.M.T.  isfoundas  follows  : 

Gr.  Sid.  T.  =  5h  36m  26s  +  4h  44m  30s •    .     =  10h  20m  568 

March  29,  Gr.  Sid.  T.  of  Gr.  M.  noon      .....  0    30    37.57 

Hence  the  sidereal  interval  elapsed  since  Gr.  M.  noon  is      9    50    18  .43 
And,  by  Table  II,  the  quantity  to  be  subtracted  from 

this  to  give  the  equivalent  mean  interval  is        ...  1    36  .71 

Hence  the  corresponding  mean  time  interval  is    .     .     .      9   48    41 .72 

This  is  the  mean  time  interval  since  Greenwich  mean  noon,  which 
of  course  is  the  required  G.M.T. 

We  may  now  determine  the  sun's  place  at  9h  48m,  or  9h.8,  G.M.T., 
by  means  of  page  II  of  the  Ephemeris,  as  follows  : 

SUN'S  R.A.  DECL.  H.  DIFFS. 

At  Gr.  M.  noon     .     .     .        ,0h  31m  33*  +  3°  24'.4  -f  9s.l         +  58" 

Diff.  for  9h.8     ....     +          1     29  +        9 .5  x  9.8         x    9  .8 

At  time  of  observation    .          0    33       2  -f  3   34  .9  89  568 

Observed  values  (p.  67)  .         0    33     19  +  3    36  lm  29s  9'.  5 


THE  NAUTICAL  ALMANAC 


107 


Determination  of  the  Equinox.  —  The  following  data  from  the  Alma- 
nacs of  1899  and  1900  may  be  compared  with  the  results  of  page  89  : 


AT  GK. 

APP.  NOON 

SUN'S  DECL. 

DJFF. 

DATE  OF  EQUINOX  BY 

INTERPOLATION 

1899. 
1900. 

Sept.  22.0 
23.0 

Sept.  23.0 
24.0 

+  0°18/    8".  7 
-0     5  13  .9 

+  0     0  26  .4 
-0   22  57  .5 

23'22".6 
23  23  .9 

23'  22".6 
0'26".4  , 

=  Sept.  23.01880 

23'  23".  9 

The  longitude   of  the  place  of  observation  was  4h  48m  408  W. 

4h  48m  40s         4.811H 


24h 


24 


days  =  Od.20046. 


Hence  the  local  dates  of  the  equinoxes  were  September  22.57574, 
1899,  and  September  22.81834,  1900,  and  the  length  of  the  tropical 
year  was  365.24260  days,  as  compared  with  the  observed  values 

September  22.714,  1899, 

September  22.856,  1900. 
365.14  days. 

Observations  of  Star  Places.  — The  right  ascensions  and  declinations 
of  the  stars  given  on  pages  86  and  89  may  be  compared  with  the 
mean  places  given  in  the  Nautical  Almanac  for  1899  and  1900,  or, 
better,  with  the  apparent  places  given  in  Part  II  of  the  American 
Ephemeris.  From  the  latter  we  find  for  September  22,  1900 : 


E.A. 

77  Serpentis 18h  16m  118.4 

X  Sagittarii 18     21     51  .9 

Vega "...     18    33     35.5 

Altair     .  19    45     57.9 


DECL. 

-  2°  55'.  3 

-  25    28  .5 
+  38   41  .8 
+    8    36.6 


which  are  in  close  agreement  with  the  results  of  observation. 


CHAPTER  VIII 
THE  CELESTIAL  GLOBE 

WHEN  a  globe  such  as  that  described  on  page  63  has  had  a  num- 
ber of  constellations  plotted  on  it  in  their  proper  positions,  and  the 
sun's  path  added,  showing  the  positions  occupied  by  the  sun  at  dif- 
ferent times  of  the  year,  it  becomes  a  very  useful  apparatus  for 
many  purposes. 

If,  for  instance,  it  is  so  placed  that  its  axis  points  to  the  pole, 
and  is  turned  about  the  axis  until  the  place  of  the  sun  as  marked 
on  the  globe  for  a  certain  date  is  on  the  under  side  and  in  a  vertical 
plane  through  the  center,  the  sphere  will  represent  the  heavens  as 
seen  at  midnight  on  the  given  date. 

When  the  globe  has  been  so  adjusted,  if  a  straight  line  is  drawn 
from  the  center  to  any  star  on  the  surface  of  the  globe,  the  prolon- 
gation of  this  line  will  lead  to  the  real  star  at  the  point  which  it 
occupies  on  the  sphere  of  the  heavens.  Thus  used,  such  a  globe  is 
helpful  to  a  beginner  in  identifying  the  constellations.  Obviously 
the  plane  of  the  sun's  path  on  the  globe,  if  extended  to  the  heavens, 
will  mark  out  the  ecliptic,  and  all  the  hour-circles  and  parallels  of 
declination  will  mark  the  corresponding  circles  in  the  sky. 

If  the  globe  is  turned  slowly  about  its  axis  so  that  a  point  on 
the  equator  moves  from  east  to  west  through  15°  per  hour,  we  have 
a  sort  of  working  model  of  the  moving  sphere  of  the  heavens  on 
which  we  may  measure  off  arcs  and  angles  and  thus  solve  approxi- 
mately many  problems  suggesting  themselves  to  one  beginning  to 
study  the  apparent  motions  of  the  heavens.  Such  an  apparatus  has 
from  very  early  times  been  an  important  aid  to  astronomers  and 
students  of  astronomy,  and  no  aid  is  so  useful  in  arriving  easily  at 
correct  ideas  on  the  subject.  Especially  was  it  useful  and  appropri- 
ate in  those  days  when  the  mechanism  of  the  heavens  was  believed 
to  correspond  closely  to  that  of  the  model  and  the  globe  was  regarded 
as  being  a  fair  representation  of  their  actual  construction,  —  in  fact, 

108 


THE  CELESTIAL  GLOBE  109 

a  representation  of  the  eighth  or  outer  sphere  which  carried  the 
fixed  stars,  turning  about  a  material  axis  somehow  fixed  in  the 
"  Primum  Mobile."  The  planets  moving  inside,  each  in  its  crystal 
sphere,  were  treated  by  projecting  them  each  on  to  its  proper  place 
on  the  outside  sphere  for  any  particular  time  to  solve  a  given  prob- 
lem. For  the  beginner,  who  stands  to  a  certain  extent  in  the  place 
of  the  early  astronomers,  it  is  still  most  important  in  studying  many 
problems.  Usually  the  diagrams  by  which  we  illustrate  our  state- 
ments of  astronomical  problems  are  drawn  as  if  the  celestial  sphere 
were  seen  from  the  outside  as  we  see  the  globe.  This  is  because 
it  is  impossible  to  represent  on  a  plane  any  large  part  of  a  spherical 
surface  as  seen  from  the  inside. 

As  usually  constructed  for  demonstration  and  the  solution  of 
problems,  the  celestial  globe  is  made  by  building  up  layers  of  strong 
paper  laid  in  glue  upon  a  solid  wooden  sphere  so  as  to  cover  it  with  a 
light  but  stiff  shell,  which  is  then  cut  through  along  a  great  circle, 
so  that  the  core  may  be  taken  out.  The  two  halves  of  the  shell 
are  fastened  together  by  gluing  on  a  strip  of  thin,  strong  cloth, 
and  after  passing  an  axis  of  stiff  wire  through  the  center,  several 
layers  of  a  mixture  of  glue  and  whiting  are  applied  to  the  surface, 
each  being  smoothed  before  drying.  The  whole  is  then  turned  so 
as  to  form  a  very  light  and  accurate  spherical  shell.  Upon  the 
surface  are  pasted  gores  of  paper,  on  which  the  circles  and  principal 
stars  are  printed  in  such  a  manner  as  to  lie  in  their  proper  places 
on  the  globe.  The  outlines  of  the  constellations  are  shown  on  the 
plates,  and  the  conventional  figures  which  have  been  ascribed  to 
them.  A  small  circular  piece  centered  on  the  pole  completes  the 
map.  The  figures  are  colored  by  hand,  and  the  whole  is  then  cov- 
ered with  a  hard,  transparent  varnish. 

Both  equinoctial  and  ecliptic  are  graduated  to  degrees,  and  the 
hours  of  right  ascension  on  the  former  are  marked  by  Koman 
numerals,  '  The  places  of  the  sun  are  usually  indicated  on  the 
ecliptic  at  dates  five  days  apart.  Since  the  circuit  of  the  sun  is 
completed  in  365J  days,  while  the  length  of  the  year  is  sometimes 
365  and  sometimes  366  days,  an  average  position  of  the  sun  must  be 
chosen,  which  is  done  with  sufficient  accuracy  by  plotting  its  place 
for  the  second  year  after  leap  year. 


110  LABORATORY  ASTRONOMY 

The  axis  of  the  globe  is  supported  by  a  stiff  brass  circle,  so  that 
the  center  of  the  sphere  lies  exactly  in  the  plane  of  one  of  its 
faces,  and  this  face  is  graduated  into  degrees,  one  semicircle  near 
the  outer  edge  from  0°  at  either  pole  to  90°  at  the  equator,  and  the 
other  semicircle  near  the  inner  edge  from  0°  at  the  equator  to  90° 
at  either  pole.  The  inner  graduation  is  used  for  measuring  the 
angular  distance  from  the  equator  to  any  point  on  the  globe,  that  is, 
the  declination  of  any  point.  The  graduation  on  the  outer  edge 
is  used  for  placing  the  axis  at  the  proper  angle  to  the  horizon  in 
rectifying  the  globe,  as  explained  on  page  111.  This  graduated 
circle  which  supports  the  axis  is  called  the  "  brass  meridian."  It 
is  mounted  in  two  slots  in  a  somewhat  larger  wooden  circle  called 
the  "horizon,"  in  such  a  manner  that  it  is  perpendicular  to  the 
latter  and  that  its  center  lies  in  the  plane  of  the  upper  surface  of 
the  wooden  circle. 

The  horizon  is  graduated  on  its  inner  edge,  and  each  quadrant 
has  two  sets  of  numbers,  one  of  which  reads  from  0°  at  the  prime 
vertical  to  90°  at  the  meridian,  and  the  other  from  0°  at  the 
meridian  to  90°  at  the  prime  vertical.  These  numbers  serve  for 
the  direct  reading  of  amplitude  and  bearing  respectively,  which  are 
easily  translated  into  azimuth,  remembering  that  W.  is  90°,  N.  180°, 
and  E.  270°,  if  azimuth  is  measured  from  the  south  point  toward 
the  west  from  0°  to  360°.  The  brass  meridian  may  be  turned  in  its 
own  plane,  sliding  easily  in  the  slots  so  that  the  axis  of  the  globe 
shall  make  any  desired  angle  with  the  horizon. 

If  the  globe  is  accurately  made  and  mounted,  its  center  will  coin- 
cide with  the  common  center  of  the  graduated  face  of  the  brass 
meridian  and  the  upper  surface  of  the  horizon,  whatever  may 
be  the  inclination  of  the  axis.  No  irregularities  should  appear 
in  the  small  space  between  these  circles  and  the  surface  of  the 
globe  when  the  latter  is  whirled  rapidly  on  its  axis.  Some  idea 
of  the^correct  placing  of  the  circles  on  the  globe  may  be  obtained 
by  noting  whether  all  points  of  the  equator  and  parallels  come 
under  the  proper  divisions  of  the  brass  meridian,  whether  all  points 
of  the  equator  pass  through  the  east  and  west  points  of  the  horizon 
90°  from  the  graduated  face  of  the  brass  meridian,  and  whether 
the  points  of  the  equator  which  lie  in  the  east  and  west  points  of 


THE  CELESTIAL  GLOBE  HI 

the  horizon  are  twelve  hours  apart  whatever  the  inclination  of 
the  axis. 

It  is  desirable  to  have  a  means  of  fixing  a  point  on  the  globe  by 
some  mark  that  may  be  afterward  removed  without  injuring  the  sur- 
face. Gummed  paper  should  not  be  used  :  small  pieces  of  unglazed 
paper  when  well  moistened  will  adhere  long  enough  for  ordinary 
purposes. 

A  good  mark  may  be  made  with  water-color  paint  mixed  with 
glycerine  so  as  to  be  very  thick  and  applied  with  a  rubber  point  or 
soft  pen  point.  Such  a  mark  may  easily  be  removed  with  a  moist- 
ened finger  even  after  several  weeks. 

Ink  suitable  for  fountain  pens  is  usually  safe  if  removed  within 
an  hour  or  two. 

TO   RECTIFY  THE   GLOBE 

In  order  that  the  globe  shall  represent  the  heavens  at  any  partic- 
ular place,  the  axis  must  be  inclined  to  the  horizon  by  an  angle 
equal  to  the  latitude.  This  may  be  accomplished  by  rotating  the 
brass  meridian  in  its  plane  and  measuring  the  angle  of  elevation  of 
the  pole  by  the  outside  graduation,  which  reads  from  0°  at  the  pole 
to  90°  at  the  equator.  This  process  is  called  "  rectifying  "  the  globe 
for  a  given  place. 

Having  been  rectified  for  a  given  place,  the  globe  may  be  rectified 
for  a  given  time  by  bringing  it  to  such  a  position  that  a  line  drawn 
from  its  center  to  any  star  is  parallel  to  the  line  drawn  from  the 
given  place  to  the  actual  place  of  the  star  in  the  heavens  at  the 
given  time.  For  this  purpose,  the  pole  being  elevated  to  the  proper 
inclination,  that  is,  the  latitude,  the  whole  apparatus  is  turned  on 
its  base  until  the  brass  meridian  is  in  the  meridian  of  the  place, 
and  the  globe  is  turned  on  the  polar  axis  until  some  one  point  is 
known  to  be  in  the  proper  position ;  then  all  points  of  the  globe 
will  be  in  their  proper  positions. 

The  point  chosen  for  this  purpose  will  vary  with  circumstances. 
If  the  local  sidereal  time  is  given,  it  is  only  necessary  to  place  the 
globe  so  that  the  hour-angle  of  the  vernal  equinox  equals  the  given 
sidereal  time.  (See  page  57.)  This  is  easily  done  by  the  graduation 


112  LABORATORY  ASTRONOMY 

of  the  equator  on  the  globe.  When  the  hour-angle  of  the  vernal 
equinox  is  lh,  2h,  3h,  the  reading  of  the  equinoctial  under  the  brass 
meridian  is  lh,  2h,  3h,  etc.,  and  the  globe  is  therefore  rectified  to  a 
given  sidereal  time  by  turning  it  about  the  polar  axis  until  the  given 
sidereal  time  is  brought  to  the  graduated  face  of  the  brass  meridian. 
The  vernal  equinox  will  then  be  at  the  proper  hour-angle  and  all 
points  on  the  globe  will  be  properly  related  to  the  corresponding 
points  on  the  sky. 

If  the  apparent  time  is  given,  the  globe  may  be  rectified  by  the 
following  process.  Mark  the  place  of  the  sun  in  the  ecliptic  for 
the  given  day.  Bring  this  point  to  the  meridian,  which  rectifies 
the  globe  for  apparent  noon ;  then,  to  rectify  it  for  the  given  ap- 
parent time,  it  is  necessary  to  turn  the  globe  until  the  hour-angle 
of  the  sun  is  equal  to  the  given  apparent  time.  This  may  be  done 
by  using  the  graduations  of  the  equator  as  follows.  Rectify  for 
apparent  noon  and  read  the  hours  and  minutes  of  the  graduation 
on  the  equinoctial  which  comes  under  the  brass  meridian  (this  is 
the  sidereal  time  of  apparent  noon).  Add  to  this  reading  the  given 
apparent  time,  and  the  sum  will  be  the  hours  and  minutes  of  the 
equatorial  graduation  that  must  be  brought  to  the  meridian  to 
place  the  sun  at  the  proper  hour-angle. 

If  local  mean  time  is  given,  the  apparent  time  may  be  obtained 
by  applying  the  correction  for  the  equation  of  time  for  the  given 
date,  and  the  globe  may  then  be  rectified  for  apparent  time,  as 
described  in  the  last  paragraph. 

If,  as  will  generally  be  the  case,  standard  time  is  given,  this  may 
be  reduced  to  local  mean  time  by  applying  the  correction  for  longi- 
tude, and  we  may  then  proceed  as  before. 

We  may  here  remark  that  in  rectifying  the  globe  for  solar  time 
we  make  use  of  the  sun's  place  as  marked  on  the  ecliptic  for  the 
given  date ;  and  that  this  place  may  be  inaccurate  by  as  much  as 
half  a  degree  is  obvious  from  the  following  consideration.  Suppose 
the  place  of  the  sun  on  the  globe  to  be  exact  for  any  one  year  on 
February  28.  It  will  be  exact  on  March  1  or  about  1°  in  error, 
according  as  the  year  has  not  or  has  the  date  February  29.  The 
following  table  of  the  sun's  longitude  shows  more  clearly  the 
nature  of  the  facts. 


THE   CELESTIAL  GLOBE 


113 


YEAR 

FEBRUARY  20 

MARCH  2 

SEPTEMBER  23 

1901    

331°.  2 

34P.2 

179°.  8 

1902   

330  .9 

341  .0 

179  .5 

1903   .     .  -V         .     .     . 

330   7 

340   7 

179  3 

1904   

330  .5 

341  .5 

180  .0 

Average          

330  .8 

341  .1 

179  .7 

The  values  nearly  repeat  themselves  after  four  years. 

It  is  obvious  that  by  assuming  an  average  value  of  the  longitude 
for  February  20,  March  2,  and  September  23,  we  should  sometimes 
be  in  error  by  about  £°  in.  the  sun's  place,  though  never  more,  and 
by  some  such  compromise  the  places  must  be  selected  for  the  posi- 
tion of  the  sun  upon  a  globe  for  general  use.  The  error  that  thus 
arises  may  amount  to  2m  in  the  determination  of  the  sun's  right 
ascension  from  the  globe. 

An  indispensable  attachment  for  the  celestial  globe  is  a  thin 
flexible  strip  of  brass  graduated  to  degrees  and  so  constructed  that 
it  may  be  attached  to  the  brass  meridian  at  its  highest  point  by  a 
pivot,  about  which  it  can  be  turned  so  as  to  be  brought  to  coincide 
with  any  vertical  circle ;  its  graduated  edge  may  then  be  brought 
over  any  point 'on  the  globe  and  the  azimuth  of  the  point  fixed  by 
noting  the  place  where  the  arc  meets  the  graduations  on  the  horizon. 
The  altitude  of  the  point  may  be  directly  read  on  the  flexible  arc, 
which  is  graduated  from  0°  at  the  horizon  to  90°  at  the  place  where 
it  is  fixed  to  the  brass  meridian.  The  graduations  are  continued 
below  the  horizon  from  0°  to  18°  for  the  purpose  of  determining 
the  end  of  twilight  (page  133).  The  flexible  arc  is  usually  called 
the  "altitude  arc." 

The  globe  thus  equipped  may  be  used  for  the  approximate  solu- 
tion of  all  problems  which  arise  from  the  diurnal  motion,  some  of 
which  we  will  now  discuss.  These  approximate  solutions  are  not 
only  sufficient  for  many  purposes,  but  always  indicate  the  proper 
statement  of  the  problem  for  purposes  of  computation,  and  serve 
to  detect  gross  errors  in  the  numerical  results. 


114 


LABORATORY  ASTRONOMY 


PROBLEMS  WHICH  DO  NOT  REQUIRE  RECTIFICATION 
OF  THE   GLOBE 

Many  problems  are  independent  of  the  position  of  the  observer 
on  the  earth's  surface,  and  for  their  solution  it  is  immaterial  at 
what  angle  the  polar  axis  is  inclined.  By  bringing  the  axis  to  the 
plane  of  the  horizon,  any  star  may  be  brought  to  view  above  the 
horizon,  but  unless  it  is  convenient  to  stand  so  that  one  can  look 
down  upon  the  globe  from  above,  it  is  often  better  to  take  a  sitting 
position  and  place  the  polar  axis  nearly  vertical.  In  following  the 
solutions  of  the  examples  below,  the  accompanying  figures  serve  to 
show  whether  the  globe  has  been  brought  to  the  proper  position. 
Problem  1.  —  To  find  the  right  ascension  and  declination  of  a  star. 
Kotate  the  globe  until  the.  star  is  in  the  plane  of  the  brass 
meridian  ;  note  the  hours,  minutes,  and  seconds  of  that  graduation 
of  the  equinoctial  which  falls  under  the  brass  meridian.  This  is  the 
right  ascension  of  the  star.  This  value  we 
may  call  the  "meridian  reading"  of  the 
equator  and  in  future  abbreviate  to  R.A.M. 
(right  ascension  of  the  meridian).  The 
declination  of  the  star  equals  that  degree 
of  the  graduation  of  the  meridian  under 
which  the  star  lies. 

Example  1.  The  star  rj  Ursse  Majoris  in 
the  end  of  the  Dipper  handle  is  brought  to 
the  brass  meridian  (Fig.  42)  and  is  found 
to  lie  halfway  between  the  divisions  49  and 
50  north  of  the  equator ;  the  declination  is 
therefore  +  49°.5.  The  meridian  reading 
is  13h  44m,  which  is  the  star's  right  ascen- 
sion. (For  reading  the  declination  the  graduations  on  the  inner 
edge  of  the  brass  meridian  must  be  used.) 

Problem  2.  —  Given  the  right  ascension  and  declination  of  a  star, 
to  find  the  star. 

Rotate  the  globe  until  the  meridian  reading  (R.A.M.)  is  equal  to 
the  given  right  ascension,  and  under  the  brass  meridian  at  the 
given  declination  will  be  found  the  star. 


FIG.  42.    R.A.M.  13"  44'° 
Decl. 


THE  CELESTIAL  GLOBE 


115 


FIG.  43.    K.A.M.  19^  46°> ; 
Decl.  +  8£° 


Example  2.  The  right  ascension  of  a  certain  star  is  19h  46m  and 

its  declination  +  8jL°.     What  is  the  star  ? 

The  division  on  the  equator  marked  19h  46m  is  brought  to  the 

brass  meridian  (Fig.  43),  and  halfway  between  the  graduations  8 

and  9  on  the  meridian  is  found  Altair,  which 

is  the  star  sought. 

Problem  3.  —  To  find  the  angular  distance 

between  two  stars. 

Place  the  flexible  quadrant  along  the  sur- 
face of  the  globe  so  that  its  graduated  edge 

passes   through   both    stars,  and  read  the 

graduation  at  the  points  where  it  touches 

each  star ;  the  difference  of  the  readings  is 

the  angular  distance  between  the  stars.  The 

graduated  edge  should  lie  along  the  great 

circle;  as  this  is  not  always  easy  to  adjust, 

it  is  well  to  repeat  the  measure  with  the 

quadrant  in  different  adjustments  and  take 

the  smallest  value  obtained.    An  alternative  method  free  from  this 

source  of  error  is  to  adjust  the  points  of  a  pair  of  compasses  so  that 

they  may  just  span  the  distance  between  the  two  stars.  The  com- 
passes may  then  be  applied  to  the  globe 
with  one  leg  at  the  vernal  equinox  (0°);  the 
other  leg  being  brought  to  the  equinoctial 
its  reading  will  give  the  angular  distance 
between  the  stars.  To  guard  against  defects 
in  the  globe,  the  second  point  may  be 
brought  to  the  ecliptic,  and  the  reading 
should  be  the  same  as  on  the  equinoctial  ; 
if  the  readings  differ,  the  mean  of  the  values 
should  be  taken. 

In  the  use  of  the  compasses  care  must 
be  taken  not  to  scratch  the  surface  of  the 

FIG.  44.    Length  of  Dipper  26°    g^^ 

Example  3.  The  following  measures  were  made  to  determine  the 
distance  between  allrsse  Majoris  and  TjUrsse  Majoris.  With  the 
flexible  quadrant  applied  to  the  globe  (Fig.  44)  so  as  to  lie  as  nearly 


116  LABORATORY  ASTRONOMY 

as  possible  along  the  great  circle  between  the  stars,  the  readings 

were: 

ij  URS.E  MAJOKIS         a  UKS^E  MAJOBIS  DISTANCE 

0.0  26.0  26.0 

0.0  26.1  26.1 

20.0  46.1  26.1 

40.0  66.1  26.1 

Here  no  difficulty  was  found  in  laying  the  arc  along  the  great 
circle,  as  the  distance  is  not  great,  and  the  value  is  taken  to  be 
26°.l.  Adjusting  the  points  of  a  pair  of  compasses  to  the  stars 
and  then  placing  the  compasses  with  one  point  at  the  vernal  equi- 
nox, the  other  point  was  found  to  reach  to  25°. 6  of  right  ascension 
on  the  equinoctial  and  to  25°.6  of  longitude  on  the  ecliptic,  which 
gives  the  distance  between  the  stars  as  25°.6. 

Problem  4.  —  To  find  the  sun's  longitude,  rig/it  ascension,  and 
.declination  at  a  given  date. 

If  the  sun's  place  at  different  dates  is  marked  on  the  ecliptic, 
its  longitude  may  be  read  off  directly  on  the  graduations  of  the 
ecliptic.  In  all  old  globes,  however,  and  in  many  modern  ones  the 
ecliptic  is  not  thus  marked,  and  the  place  of  the  sun  must  be 
found  by  determining  the  longitude  by  a  table  such  as  that  given 
on  page  173,  which  is  nearly  correct  for  the  first  half  of  the  present 
century.  A  substitute  for  this  table  is  generally  to  be  found  in 
the  form  of  two  contiguous  concentric  circles  on  the  horizon  circle, 
one  graduated  into  degrees  of  longitude  and  the  other  into  months 
and  days,  so  that  the  line  for  a  given  date  in  the  outer  circle  is 
found  opposite  the  corresponding  degree  of  the  sun's  longitude  in 
the  inner  circle.  Commonly  also  the  divisions  both  of  this  circle 
and  of  the  ecliptic  are  divided  into  groups  of  30°,  each  correspond- 
ing roughly  to  one  month  of  time.  The  30°  of  Aries  reach  from  the 
first  of  Aries  on  March  20  to  the  first  of  Taurus  on  April  20,  and  so 
on  in  the  order  of  the  signs.  Thus,  opposite  May  6  is  the  fifteenth 
degree  of  Taurus,  corresponding  to  longitude  45°  in  the  usual  way 
of  reckoning ;  opposite  January  1  is  the  tenth  degree  of  Capricor- 
nus,  nine  complete  signs  and  10°,  or  longitude  280°.  In  the  table 
on  page  173  the  equivalents  of  the  degrees  of  longitude  are  given 
in  signs  and  degrees. 


THE  CELESTIAL  GLOBE  117 

By  whatever  method  the  sun's  place  in  the  ecliptic  is  fixed, 
its  right  ascension  and  declination  are  found  by  the  method  of 
Problem  1. 

Example  4-  What  are  the  sun's  right  ascension  and  declination 
on  April  20  ? 

The  longitude  is  found  by  the  table  to  be  29°.5,  and  on  bringing 
this  point  of  the  ecliptic  to  the  meridian  (Fig.  45)  it  is  found  to  be 
in  declination  -f-11^-0,  while  the  reading  of 
the  meridian  is  lh  50m.  The  sun's  right 
ascension  is  therefore  lh  50m  and  its  decli- 
nation is  11-°  north. 


PROBLEMS  WHICH  REQUIRE  RECTI- 
FICATION OF  THE  GLOBE  FOR  A 
GIVEN  TIME 

Such  are  problems  which  require  a  deter- 
mination of  the  angle  between  the  meridian 
and  some  one  of   the  hour-circles  of   the     Fl(>-  45.   Sun's  E.A.  i*  53m ; 
globe.    They  are  independent  of  the  latitude 

of  the  place  of  observation,  but  depend  upon  the  position  of  the 
heavenly  bodies  with  respect  to  the  meridian.  The  brass  meridian 
being  taken  as  the  meridian  of  the  place  of  observation,  the  only 
quantities  involved  are  differences  of  hour-angle  and  of  right 
ascension,  and  it  will  be  advisable  here  to  collect  the  following  rela- 
tions, which  have  already  been  explained. 

All  time  is  measured  by  the  continually  increasing  hour-angle  of 
some  point  of  the  celestial  sphere. 

Local  sidereal  time  (Camb.  Sid.  T.)  is  the  hour-angle  of  the  vernal 
equinox. 

Local  apparent  (solar)  time  (Camb.  App.  T.)  is  the  hour-angle 
of  the  sun. 

Local  mean  (solar)  time  (Camb.  M.  T.)  is  the  hour-angle  of  the 
mean  sun. 

For  example,  at  21h  20m,  Camb.  Sid.  T.,  the  hour-angle  of  the 
vernal  equinox  at  Cambridge  is  21h  20m ;  at  10h  30m,  Chicago  apparent 


118  LABORATORY  ASTRONOMY 

time,  the  hour-angle  of  the  sun  at  Chicago  is  10h  30m ;  at  5h  10m, 
New  York  mean  time,  the  hour-angle  of  the  mean  sun  at  New  York 
is  5h  10m. 

The  hour-angle  is  in  all  cases  measured  westward  from  the 
observer's  meridian  up  to  24h. 

Greenwich  mean  time  (G.M.T.)  is  the  hour-angle  of  the  mean 
sun  measured  from  the  meridian  of  Greenwich.  When  we  say 
that  a  place  is  a  certain  number  of  hours  and  minutes  of  longitude 
west  of  Greenwich,  we  mean  that  the  rotation  of  the  earth  brings 
the  sun  to  the  meridian  of  the  place  just  so  many  hours  and  minutes 
after  its  arrival  at  the  meridian  of  Greenwich.  At  local  noon,  then, 
its  hour-angle,  reckoned  from  the  Greenwich  meridian,  is  equal  to 
the  difference  of  longitude  between  the  two  meridians.  As  the  sun 
thereafter  moves  westward  equally  from  the  two  meridians,  Green- 
wich time  is  always  greater  than  that  of  any  place  west  of  it  by 
exactly  the  difference  of  their  longitudes. 

Therefore,  to  find  the  G.M.T.  corresponding  to  a  given  local  mean 
time,  we  add  to  the  latter  the  longitude  (west)  from  Greenwich. 
Standard  time  is  directly  obtained  from  G.M.T.  by  subtracting  4, 
5,  6,  7,  8  hours,  respectively,  for  Colonial,  Eastern,  Central,  Moun- 
tain, and  Pacific  time.  Thus,  the  "reduction  for  longitude,"  so 
called,  from  Cambridge  mean  time  is  +  4h  44m.5  to  G.M.T.  and 
-f-  4h  44m.5  —  5h  to  Eastern  standard  time  ; 
or,  by  a  single  operation,  —  15m.5  directly  to 
Eastern  time.  The  "  reduction  for  longi- 
tude" for  San  Francisco  is  -f  8h  9m.7  to 
Greenwich  and  +  8h  9m.7  -  8h  =  +  9m.7  to 
Pacific  time.  Problems,  therefore,  which 
involve  standard  time  require  a  knowledge 
of  the  observer's  longitude. 

Problem  5.  —  To  rectify   the  globe  for  a 
given  sidereal  time. 

Eotate  the  globe  till  the  E.A.M.  equals 
the  given  sidereal  time.     This  brings  the 
FIG.  46.   Sid.  T.  7"  50-        vernal  equinox  to  an  hour-angle  equal  to 
the  given  sidereal  time,  and  all  points  of  the  sphere  into  their 
proper  relation  to  the  meridian. 


THE  CELESTIAL  GLOBE  119 

Example  5.  To  rectify  the  globe  for  7h  50m  sidereal  time,  rotate 
the  globe  until  E.A.M.  is  7h  50m  (Fig.  46). 

Problem  6.  —  The  globe  being  rectified  for  a  given  sidereal  time, 
to  determine  the  hour  angle  of  a  body. 

Note  the  E.A.M.  when  the  globe  is  in  the  given  position;  then 
bring  the  body  to  the  meridian  and  read  its  right  ascension.  Sub- 
tract the  latter  reading  from  the  former  and  the  result  is  the  hour- 
angle  of  the  body. 

Since  the  reading  of  the  meridian  is  always  the  sidereal  time 
(page  59),  this  process  exemplifies  the  equation  H.A.  =  Sid.  T. 
—  E.A.  It  is  of  course  understood .  that  if  in  adding  two  times 
or  hour-angles  the  result  is  greater  than  twenty-four  hours,  that 
amount  is  to  be  subtracted ;  thus,  an  hour-angle  of  35h  25m  108 
corresponds  to  the  same  position  as  an  hour-angle  of  llh  25m  10s. 
Also,  if  it  is  required  to  subtract  a  larger  froni  a  smaller  hour- 
angle,  the  latter  should  be  increased  by  twenty-four  hours  before 
performing  the  subtraction  :  thus,  6h  41m 
-  llh  17m  =  30h  41m  -  llh  17m  =  19h  24m. 

Example  6.  What  is  the  hour-angle  of 
Sirius  at  (a)  7h  50ra,  sidereal  time,  and  at 
(b)  4h  20m,  sidereal  time  ? 

(a)  Rectifying  the  globe,  as  in  Problem 
5,  to  7h  50m  Sid.  T.,  the  E.A.M.  =  7h  50m. 
Bringing  Sirius  to  the  meridian  (Fig.  47), 
E.A.M.  =  6h  41m  =  E.A.  of  Sirius,  as   in 
Problem    1.      Hence    H.A.    of    Sirius    at 
7h  50m  Sid.  T.  =  7h  50m  -  6h  41m  =  lh  9m 
(Fig.  46). 

(b)  Eectifying  to  4h  20m  Sid.  T.,  E.A.M.      FlG" 47'  KA'  of  Siriu8' 6" 41m 
=  4h  20m,  and,   as  before,  H.A.  =  4h  20m  -  6h  41m  =  28h  20m  - 
6h  41m  =  21h  39m. 

Problem  7.  —  The  globe  being  rectified  for  a  given  apparent  time,  to 
determine  the  hour-angle  of  a  body. 

Bring  the  sun's  place  to  the  meridian  and  take  the  E.A.M.  (this 
is  the  sun's  right  ascension,  Problem  4).  Eotate  the  globe  through 
an  hour-angle  equal  to  the  given  apparent  time,  and  the  sun  is 
brought  to  the  required  hour-angle ;  the  E.A.M.  thus  becomes  H.A. 


120  LABORATORY  ASTRONOMY 

of  the  sun  -f-  R.A.  of  the  sun,  and  the  globe  is  properly  rectified 
when  this  reading  of  the  equator  is  brought  under  the  meridian. 

Since  H.A.  +  E.A.  =  Sid.  T.,  the  rule  may  be  given  as  follows  : 
Determine  the  sun's  right  ascension  by  the  globe  (Problem  4). 
Add  the  given  apparent  time.  The  sum  is  the  sidereal  time. 
For  this  sidereal  time  rectify  the  globe  by  Problem  5,  and  find  the 
hour-angle  by  Problem  6. 

Example  7.  What  is  the  hour-angle  of  Sirius  at  10  P.M.,  apparent 
time,  February  13  ? 

Sun's  R.  A.  by  globe 21h  50m 

App.  T 10 0_ 

Sid.  T 7     50 

R.A.  of  Sirius  by  globe '      6    41 

H.A.  of  Sirius 1       9    (Fig.  46) 

Problem  8.  —  The  globe  being  rectified  for  a  given  mean  time,  to 
determine  the  hour-angle  of  a  body. 

Apply  the  equation  of  time  (with  the  proper  sign)  to  the  given 
mean  time  to  find  the  corresponding  apparent  time,  and  with  this 
value  rectify  as  in  Problem  7. 

Example  8.  What  is  the  hour-angle  of  Sirius  at  5  A.M.,  local  mean 
time,  July  10  ? 

Equation  of  time  +  5m  (add  to  App.  T.) 

July  10,  5  A.M =  July  9d  17h    Om 

Eq.  of  T.  (subtract) 5_ 

App.  T.    .     .    *** 16     55 

Sun's  R.A.  by  globe  (add)  .     .     . 7     20* 

Sid.  T 0     15 

R.A.  of  Sirius  (Problem  1)  (subtract) 6    41 

H.A.  of  Sirius 17     34 

Problem  9.  —  The  globe  being  rectified  for  a  given  standard  time,  to 
determine  the  hour-angle  of  a  body. 

Apply  the  reduction  for  longitude  to  find  the  corresponding  mean 
time  and  rectify  as  in  Problem  8. 

*  The  sun's  place  is  marked  on  the  globe  for  noon  of  the  indicated  date.  It 
is  therefore  more  accurate  in  this  problem  to  make  use  of  the  sun's  place  for 
July  10  and  in  general  for  the  nearest  noon,  which  is  always  that  of  the  civil 
date. 


THE  CELESTIAL  GLOBE 


121 


Example  9.  At  Chicago  (longitude  -f  5h  50m)  what  is  the  hour- 
angle  of  Sirius  at  6.30  P.M.,  Central  standard  time,  October  30? 

Red.  for  Long.  Chicago  T.  to  Central  T.  -  10m 

Eq.  of  T.  -  16m  (subtract  from  App.  T.) 

Central  standard  time 6h  30m 


Red.  of  Long,  to  Chicago  M.  T. 

Chicago  M.  T 

Eq.  of  T.  (add  to  M.  T.)       .     . 


+10 

6    40 

+16 

App.  T 6     56 


Sun's  R.A.  by  globe  (add) 14  23 

Chicago  Sid.  T 21  19~ 

R.A.  of  Sirius  (subtract) 6  41 

H.A.  of  Sirius  (by  Problem  5) 14  38~ 

Reduction  to  the  Equator.  —  In  the  solution  of  Example  4,  page 
117,  it  was  shown  that  when  the  sun's  longitude  is  29°.5  its  R.A. 
is  lh  50m,  or  27°.5. 

The  quantity  which  must  be  added  to  the  longitude  of  a  point 
on  the  ecliptic  to  find  its  E.A.  (in  this  case  —2°)  is  called  the 
"  reduction  to  the  equator  "  and  is  used  in  finding  the  equation  of 
time  as  explained  in  Chapter  X.  Its  value  for  any  given  point  of 
the  ecliptic  may  be  found  by  the  globe  as  in  Example  4. 

Following  are  the  results  : 


LONGITUDE 

0°  and  180° 
10     190 
20     200 
30     210 
40     220 


50 
60 
70 
80 
90 


230 
240 
250 
260 
270 


RED.  TO 
EQUATOR 

0°.0 
-0.8 
-1  .5 
-2  .1 
-2  .4 

-2  .4 
-2  .2 
-  1  .6 
-0  .9 
0.0 


LONGITUDE 

RED.  TO 
EQUATOR 

90°  and 

270° 

o°.o 

100 

280 

+  0.9 

110 

290 

+  1.6 

120 

300 

+  2  .2 

130 

310 

+  2.4 

140 

320 

+  2  .4 

150 

330 

+  2.1 

160 

340 

+  1.5 

170 

350 

+  0.8 

180 

360 

0.0 

CHAPTER  IX 


EXAMPLES    OF   THE   USE   OF  THE   GLOBE 

MOST  of  the  problems  with  which  we  have  to  deal  require  that 
the  observer's  exact  place  on  the  earth  shall  be  known,  —  that  is, 
his  latitude  as  well  as  his  longitude  ;  and  in  order  that  they  may 
be  solved  it  is  necessary  that  the  globe  should  be  rectified  to  the 
latitude  by  inclining  the  axis  to  the  horizon  by  an  angle  equal  to 
the  latitude. 

This  chapter  contains  some  typical  examples  and  the  methods 
by  which  they  are  solved,  with  references  to  the  problems  of  the 
preceding  chapter.*  Attention  should  be  paid  to  the  arrangement 
of  the  solutions,  and  all  numerical  results  should  be  fully  labeled 
so  that  it  may  be  seen  how  they  are  obtained  and  combined.  In  all 
the  problems,  unless  otherwise  stated,  the  globe  must  be  rectified  to 
the  latitude  of  Cambridge,  42°.4  N.  The  longitude  may  be  assumed 
411  4^m  wesf;  of  Greenwich. 

Example  10.  At  what  sidereal  time  do 
the  Pleiades  rise  at  Cambridge  ? 

Rectify  the  globe  by  raising  the  north 
pole  to  such  an  angle  that  the  graduation 
42°.4  on  the  outside  edge  of  the  brass  merid- 
ian coincides  with  the  surface  of  the  hori- 
zon. Rotate  the  globe  about  the  polar  axis 
until  the  Pleiades  are  in  the  plane  of  the 
eastern  horizon  (Fig.  48).  The  R.A.M. 
equals  the  sidereal  time  sought,  —  20h  12m. 
This  result  is  independent  of  the  longitude. 
The  Pleiades  rise  at  any  place  in  latitude 
42°.4  K  at  20h  12m  of  local  sidereal  time. 


FIG.  48.    Rising  of  Pleiades  : 
20h  12™  Camb.  Sid.  T. 


*  These  solutions  were  obtained  with  a  not  very  accurate  globe  nine  inches  in 
diameter.     Better  results  may  be  obtained  with  a  larger  globe  in  good  condition. 

122 


EXAMPLES  OF  THE  USE   OF  THE   GLOBE 


123 


Example  11.  At  what  apparent  time  do  the  Pleiades  rise  at  Cam- 
bridge on  October  30  ? 

Determine  the  sidereal  time,  as  in  the  last  example,  20h  12m. 
The  sun's  right  ascension  is  determined  to  be  14h  17m  by  bringing 
it  to  the  meridian  (Fig.  49),  as  in  Prob- 
lem 4,  and  the  relation  App.  T.  =  Sid.  T.  — 
Sun's  E.A.  gives 
20h  12m  -  14h  17m  =  5h  55m  Camb.  App.  T. 

Example  12.  At  what  Cambridge  mean 
time  do  the  Pleiades  rise  October  30  ?  Eq. 
of  T.  =  -  16m  (subtract  from  App.  T.). 

The  apparent  time  being  5h  55m  by  the 
last  example,  the  mean  time  is  5h  55m  — 
16m  =  5h  39m. 

Example  13.  At  what  Eastern  standard 
time  do  the  Pleiades  rise  at  Cambridge 

~    .    ,  nr.n 

October  30? 

The  arranement  of  the  work  is  as  follows  : 


FIG.  49.    October  30:  Sun's 


Camb.  Sid.  T.  by  globe  (Example  10)    ...     .....  20h  12m 

Sun's  R.A.  by  globe  (Problem  4)       .........  14     17 

Camb.  App.  T.  (Example  11)  ........  •    .    '.     .  ~6     55~ 

Eq.  of  T.  by  table  .......  -    .     ,     ......  -  16 


Camb.  M.  T.  (Example  12) 5    39 

Red.  to  E.  Std.  T.  -  16 


E.  Std.  T.  of  rising  of  Pleiades 5    23 

Example  14-  At  what  standard  time  do 
the  Pleiades  set  at  Cambridge  March  1  ? 

Bringing  the  Pleiades  to  the  western 
horizon,  we  have,  as  in  Example  13  : 

Camb.  Sid.  T.  by  globe  (Fig.  50)     ...  11*  15m 

Sun's  R.A.  (Problem  4) 22     50 

Camb.  App.  T :......  12    25 

Eq.  of  T.  by  table    .     .     .     .    • .  .  .     .     .  +  13 

Camb.  M.  T .     ...     .  12     38 

Red.  to  E.  Std.  T -16 

E.  Std.  T.  of  setting  of  Pleiades  March  1  12     22 

Example  15.  What  is  the  standard  time 

FIG.  50.    Pleiades  setting :          „  .  .    ~        ,     .  ,  ,_-         +  ~  n 

Sid.  T.  iib  i5m  of  sunrise  at  Cambridge  on  May  15  ? 


124 


LABORATORY  ASTRONOMY 


Mark  the  place  of  the  sun  on  the  ecliptic  for  May  15  and  bring 
this  point  to  the  plane  of  the  eastern  horizon  (Fig.  51). 

The  R.A.M.  gives  the  Camb.  Sid.  T.  by 

globe 20h  28m 

Sun's  R.A.  (Problem  4)  by  globe    .     .     .  3    28 

Camb.  App.  T T?     00~ 

Eq.  of  T.  by  table _      -4 

Camb.  M.  T 16     56 

Red.  for  Long — 16 

Std.  T.  of  sunrise  May  15 ~IQ 40~ 

Or,  May  16,  4h  40m  A.M.  But  see  the  note  to  Prob- 
lem 8.  Since  the  place  of  the  sun  was  taken  for 
May  15,  the  solution  gives  the  time  of  sunrise  for 
that  civil  date. 


FIG.  51.    Sunrise  May  15: 
Sid.  T.  20h  28"> 


Example  16.  What  is  the  azimuth  of  the 
sun  at  Cambridge  at  sunrise  June  21  ? 

The  sun's  place  for  June  21,  being  brought 
to  the  horizon  as  in  the  preceding  problem,  was  found  to  be  on  the 
division  59  of  the  graduation  which  reads  from  zero  at  the  north 
point  of  the  horizon  to  90°  at  the  east  point  (Fig.  52) ;  its  bearing, 
therefore,  is  N.  59°  E.,  and  its  azimuth 
reckoned  from  the  south  point  is  180°  + 
59°,  or  239°. 

The  graduation  on  the  inner  edge  of  the 
horizon  has  a  second  set  of  numbers  begin- 
ning with  0°  at  the  east  and  west  points 
and  running  to  90°  at  the  north  and  south 
points.  By  means  of  this  amplitudes  may 
be  directly  measured.  The  amplitude  of 
the  sun  in  this  case  was  E.  31°  N. 

Example  17.  At  Cambridge,  September  10, 
in  the  afternoon,  the  sun's  altitude  is  20°. 
What  is  its  azimuth? 

For  the  solution  of  this  problem  the  alti- 
tude arc  must  be  applied  to  the  brass 
meridian,  attaching  the  clamp  so  that  the  90°  mark  of  the  gradua- 
tion is  as  exactly  as  possible  under  the  graduation  42°.4  on  the 
inner  edge  of  the  brass  meridian ;  this  is  at  the  highest  point 


FIG.  52.  Sunrise  June  21 : 
Sun's  Bearing  N.  59°  E. ; 
Az.  239° 


EXAMPLES  OF  THE  USE  OF  THE   GLOBE 


125 


of  the  globe,  corresponding  to  the  zenith  of  the  sphere  in  latitude 

42°.4  north. 

The  longitude  of  the  sun  for  September  10  being  found,  by  the 

circles  printed  on  the  horizon  for  this  purpose,  to  be  17°.7  in  Virgo, 

or  five  signs  and  17°.7  =  167°.7,  this  point  was  brought  into  the 
southwest  quadrant  halfway  from  the  south 
to  the  west  point  and  the  altitude  arc  made 
to  pass  through  it ;  the  altitude  was  seen  to 
be  approximately  40°.  The  foot  of  the  arc 
was  then  moved  about  20°  toward  the  west 
point  and  the  sun's  place  brought  to  it ;  the 
altitude  was  now  about  30°.  The  foot  of 
the  arc  was  moved  again  about  20°  farther 
toward  the  west  point  and  the  sun's  place 
brought  to  it,  the 
sun's  altitude  being 
about  15°.  The  arc 

FIG.  53.   September  15 :  Sun's   was     now     moved 

Alt.  20° ;  Az.  77°.5  ,        .  .  . 

back  a  few  degrees 

toward  the  south  and  by  a  few  trials  a 
position  found  (Fig.  53)  such  that  the  sun's 
place  coincided  exactly  with  the  division 
marking  an  altitude  of  20° ;  the  zero  of  the 
graduated  edge  of  the  arc  was  then  halfway 
between  77°  and  78°  of  the  graduation  on 
the  inner  edge  of  the  horizon  circle.  The 
bearing  was  then  S.  77°.5  W.  and  the  azi- 
muth 77°.5. 

Example  18.  At  Cambridge  Altair  is  east 
of  the  meridian  at  an  altitude  of  30°.  Find  its  azimuth  and 
hour-angle  and  the  sidereal  time.  Bringing  the  place  of  Altair 
to  30°  on  the  flexible  arc,  as  described  in  the  last  problem,  the 
bearing  is  found  to  be  S.  73°  E.  Hence  the  azimuth  is  287°.  With 
the  same  adjustment  the  E.A.M.  is  15h  56m,  which  is  the  sidereal 
time.  By  bringing  Altair  to  the  meridian,  its  right  ascension  is 
found  to  be  19h  43m,  and,  by  Problem  5,  H.A.  =  15h  56m  -  19b  43m 
=  20h  13m. 


FIG.  54.  Alt.  of  Altair  30° : 
H.A.  20h  13-" ;  Az.  287°; 
Sid.  T.  15*  56* 


126 


LABORATORY  ASTRONOMY 


Example  19.  On  September  10,  at  Cambridge,  in  the  forenoon, 
the  sun's  altitude  is  20°.     What  is  the  local  mean  time  ? 

The  sun's  longitude  being  167°.7,  as  in 
Example  17,  its  place  is  brought  to  20°  on 
the  flexible  arc  in  the  southeast  quadrant 
(at  a  bearing  S.  78°  E.,  with  which  compare 
the  result  of  Problem  17)  and  the  problem 
solved  as  follows  : 

Sun's  forenoon  Alt.  20° 

R.A.M. 6h  42m 

Sun's  R.  A.  (Problem  4) 11     13 

App.  T 19    29 

Eq.  of  T.  by  table -3 

Camb.  M.  T 19    26 

FIG.  55.    Sun's  Alt.  20°;          Or    •      •      -  , 7      26A.M. 

K.A.M.  G»>  42«> 

It  would  appear  that  our   result  means 

7.26  A.M.  of  the  following  day.  But  it  is  to  be  remembered  that  we 
have  used  the  sun's  place  for  September  10  (the  places  are  marked 
for  noon),  and  our  solution  then  applies  more  nearly  to  the  morning 
of  that  date.  Example  19  is  perhaps  the  most  important  that  we 
have  solved,  since  it  illustrates  the  method 
by  which  the  longitude  is  determined  at 
sea.  The  sun's  altitude  is  measured  by  a 
sextant  and  its  hour-angle  computed.  From 
the  apparent  time  thus  obtained  the  local 
mean  time  is  found  as  above  and  compared 
with  G-.M.T.  kept  by  a  chronometer. 

Example  20.  On  July  10,  at  Cambridge, 
what  is  the  sun's  hour-angle  when  it  is  in 
the  prime  vertical  ?  What  is  the  local 
mean  time  ? 

In  the  summer  half  of  the  year  the  sun 
is  in  the  prime  vertical  once  in  the  fore- 
noon and  once  in  the  afternoon,  so  that 
there  will  be  two  solutions  of  the  problem. 

The  place  of  the  sun  July  10  is  found  by  the  table  to  be  in 
longitude  107°.7.     The  altitude  arc  being  adjusted  with  its  foot 


FIG.  5G.  Sun  in  Prime  Verti- 
cal :  July  10,  forenoon ; 
R.A.M.  3h  3"> 


EXAMPLES  OF  THE  USE   OF  THE   GLOBE 


127 


at  the  east  point  of  the  horizon,  the  sun's  place  is  brought  to  the 
graduated  edge  of  the  are  and  R.A.M.  noted.  The  altitude  arc 
being  brought  in  the  same  way  to  coincide  with  the  west  quadrant 
of  the  prime  vertical,  the  sun's  place  is  brought  again  to  the  gradu- 
ated edge  and  R.A.M.  noted.  Then  the  sun's  right  ascension  is 
determined,  and  the  results  may  be  recorded  and  the  computation 
made  in  the  following  form  : 


Sun  in  prime  vertical  A.M. 

R.A.M 3h    3m 

Sun's  R.A.  by  globe    .  '.     .'    .     .     .    '.     .     .  7     20 

App.  T >   .     .     .     .     .  19    43 

Eq.  of  T.  by  table +  5 

Local  M.  T.    . 


19    48 


Llh  36m 
7     20 
4     16 
+  5 
4     21 


Or 


7     48A.M.,       4     21p.M. 


Example  21.  At  Cambridge,  at  Oh  sidereal  time,  what  bright 
stars  are  seen  near  the  meridian  ?  What  are  their  declinations  ? 

Eectify  the  globe  for  latitude  +  42°.4.  Eotate  the  globe  until 
the  R.A.M.  is  Oh,  and  the  following  stars 
will  be  found  near  the  meridian  :  y  Pegasi, 
Decl.  +  14°.0  ;  a  Andromedse,  Decl.  +  27°.5; 
ft  Cassiopeise,  Decl.  +  58°;  Polaris,  of  course, 
but  too  near  the  pole  to  be  seen  on  the  globe  ; 
y  Ursse  Majoris,  Decl.  54° ;  8  Ursse  Majoris, 
Decl.  58°.  The  two  latter  are  below  the 
pole,  and  to  determine  their  declinations 
the  globe  must  be  rotated  180°  to  bring  them 
under  the  inner  graduations  of  the  meridian. 

Notice  that  the  four  first  stars  lie  along 
the  same  hour-circle,  which  is 'the  equinoc- 
tial colure,  in  R.A.  Oh,  and  that  this  circle 
is  divided  roughly  by  them  into  multiples 
of  15°,  thus  :  Polaris  to  (3  Cassiopeise,  30° ; 
ft  Cassiopeise  to  a  Andromedse,  30°;  a  Andromedse  to  y  Pegasi,  15°. 

By  continuing  the  line  of  stars  about  15°  we  arrive  at  Decl.  =  0°, 
R.A.  =  0°,  that  is  at  the  vernal  equinox,  which  though  marked  by 
no  conspicuous  star  is  easily  fixed  by  this  alignment. 


FIG.  57.  Stars  on  Meridian 
at  Cambridge  at  V*  Side- 
real Time 


128 


LABORATORY  ASTRONOMY 


Example  22.  What  is  the  standard  time  corresponding  to  Oh  of 
sidereal  time  at  Cambridge  October  10  ? 

The  sidereal  time  being  given,  this  problem  is  similar  to  Exam- 
ples 13,  14,  and  15,  and  illustrates  the  general  process  of  passing 
from  sidereal  to  mean  or  standard  time  by  means  of  the  globe,  thus  : 

Sid.  T  ..............     .....      Oh    Om 

Sun's  R.  A.  by  globe    ......    -    '  .......     13  _  4 


App.  T  ...................     10     56 

Eq.  of  T  ...............     ...         -  13 


Camb.  M.  T  .................     10    43 

Red.  for  Long,  to  Std.  T  ......    *.     .     .     .--.'.     .        -  16 


Eastern  standard  time 10     27 

Example  23.  Find  the  altitude  and  azimuth  of  Arcturus  at 
8  P.M.,  standard  time,  at  Cambridge,  September  10. 

This  problem  requires  the  globe  to  be 
rectified  for  both  latitude  and  time.  The 
latter  adjustment  is  made  as  follows : 

Std.  T 8h    Om 

Red.  for  Long +  16 

Camb.  M.  T 8      16 

Eq.  of  T.  by  table  (add  to  M.T.)      .     .     .  +3 

App.  T 8     19 

R.A.  Sun  by  globe 11     15 

Camb.  Sid.  T 19     34 

Rectify  for  Cambridge,  Lat.  +  42°.4. 
Rotate  the  globe  till  the  E.A.M.  is  19h  34m. 

FIG.  58.    Arcturus:   Septem- 

ber  10,  8  P.M.,  E.  std.  T.  ;   Apply  the  altitude  quadrant  so  as  to  pass 

Ait.  20° ;  AZ.  98°  through  Arcturus,  and  we  find  its  altitude 

19°.5,  and  its  bearing  K  80°.5  W.;  hence  its  azimuth  is  99°.5. 

Example  24.  What  constellation  is  rising  in  the  east  at  9  P.M., 
Eastern  standard  time,  at  Cambridge,  November  10? 

As  in  the  preceding  problem : 

Std.  T 9h    Om 

Red.  for  Long,  to  Camb.  M.  T.     .     .     .     , +  16 

Camb.  M.  T •  9    16 

Eq.  of  T.  by  table  (subtract  from  App.  T.) +  15 

Camb.  App.  T.       9    31 

R.A.  of  Sun  by  globe 16 3__ 

Sid.  T.  0    34 


EXAMPLES  OF  THE  USE  OF  THE  GLOBE 


129 


To  rectify  for  time  rotate  the  globe  till  the  E.A.M.  is  Oh  34m. 

It  will  be  found  that  the  constellation  of  Orion  has  just  risen 

above   the    eastern    point   of   the    horizon. 

Compare  the  form  of  this  solution  with  that 

of  Example  13,  which  is  the  inverse  of  this, 

the  rising  of  a  star  being  given  and  the 

standard  time  sought. 

PROBLEMS  INVOLVING  THE  USE  OF 
THE  NAUTICAL  ALMANAC 

Example  25.  At  Cambridge,  November  30, 
1904,  at  5h  15m  P.M.,  standard  time,  a  bright 
star  is  seen  due  southwest  about  10°  above 

,,      ,       .  ,T       ,.  ,     .  .    .,  ,     .        FIG.  59.   Orion  rising:  Cam- 

the  horizon.    .No  other  stars  being  visible  in      bridge, November  10, 9 P.M., 
the  twilight,  it  is  desired  to  identify  the  star. 

E.  Std.  T . 

Red.  for  Long.       . 

Camb.  M.T 

Eq.  of  T.  (subtract  from  A  pp.  T.)     .... 

App.  T .-    . 

R.A.  of  Sun 

Camb.  Sid.  T.   . 


Std.  T. 


5h    I5 

+  16 

5     31 

+  11 


5 

16 


42 

20 


22        8 


Rectifying  for 
Cambridge,  Lat.-f- 
42°.4,  and  for  22h 
8m  Sid.  T.,  it  is 
found,  by  means 
of  the  altitude 
arc  (Fig.  60),  that 
there  is  no  star 
upon  the  globe  at 
the  given  altitude 
and  azimuth,  the 
nearest  star  being 

FIG.  60.    Star  10°  above  South-    cr  Centauri,  which    FIG.  61.   Star  brought  to  Merid- 

re™;^^:  ™>uld  *°t  t>e  visi- 

22"  7»  ble  at  that  altitude 


ian  :  R.A  18»>  56™  ;  Decl.  - 


130  LABORATORY  ASTRONOMY 

in  twilight.  The  exact  point  being  marked  is  brought  to  the 
meridian  and  found  to  be  in  R.A.  18h  56m  and  Decl.  —  23^°  (Fig. 
61).  The  fact  that  its  position  is  very  near  the  ecliptic  suggests 
that  it  may  be  a  planet,  and  on  consulting  the  Almanac  it  is 
found  that  on  November  30  the  right  ascension  of  Venus  is  19h 
4m  and  its  declination  —  24°.7,  or  within  about  2°  of  the  observed 
place. 

Example  26.  Which  of  the  planets  that  are  visible  to  the  naked 
eye  are  above  the  horizon  at  Cambridge  at  8  P.M.,  standard  time, 
October  1,  1904  ? 

From  the  Nautical  Almanac  are  taken  the  following  data  for  the 
given  date  : 

R.A.  DECL. 

Mercury      ............     llh  25m  +    5°.l 

Venus     .............     13    55  -  11  .4 

Mars      ............     .     10    10  +  12  .7 

Jupiter  ............     .       1    44  +    9  .1 

Saturn   ...........     .     .     21    10  -  17  .0 

Marking  these  places  upon  the  globe  and 
rectifying  for  the  given  place  and  time,  it 
is  at  once  seen  that  the  first  three  are  below 
the  western  horizon,  while  Jupiter  is  20° 
above  the  east  point  of  the  horizon  and 
Saturn  approaching  the  meridian  at  an 
altitude  of  about  30°. 

Where  only  an  approximate  result  is 
desired,  it  will  often  be  sufficient  to  neglect 
the  corrections  for  longitude  and  equation 
of  time,  the  sum  of  which  at  Cambridge 
never  amounts  to  much  more  than  half  an 
FIG.  62.  pjaneta,  October  i,  hour!  Thig  of  CQurse  assumeg  standard 


time  to  equal  apparent  time.    Thus,  in  this 

problem  we  may  bring  the  sun  to  the  meridian  and,  noting  E.A.M. 
=  12h  30m  and  adding  8h,  we  have  20h  30m  (±  30m)  as  the  E.A.M. 
corresponding  to  8h  apparent  time.  The  general  terms  in  which 
the  answer  is  given  above  will  apply  equally  well,  and  some 
time  is  saved  where  only  the  general  aspect  of  the  heavens  is 
required. 


EXAMPLES  OF  THE   USE  OF  THE   GLOBE  131 

Example  27.  At  what  standard  time  does  Jupiter  set  at  Cam- 
bridge December  25,  1904? 

By  the  tables  in  the  Nautical  Almanac,  we  find  that  on  the 
given  date  the  right  ascension  of  Jupiter  is  lh  17m  and  its  declina- 
tion -f  6°.8.  Marking  this  place  on  the  globe  and  bringing  it  to 
the  western  horizon,  the  E.A.M.  is  7h  38m,  which  is  the  sidereal 
time.  Converting  to  standard  time  : 

Sid.  T 7h  38m 

Sun's  R.A.  by  globe 18  12 

App.  T 13  26 

Eq.  of  T .     .  0 

Camb.  M.  T 13  26 

Red.  for  Long -  16 

Std.  T 13  10 

Or 1  10  A.M. 

Example  28.  At  what  time  does  the  moon 
rise  at  Cambridge  December  25,  1904  ? 

If  the  moon's  position  were  known  directly 
from  the  Nautical  Almanac,  the  solution  of      pIGi  63.  jupiter  setting : 
this  problem  would  be  similar  to  the  last:         R-A-M-  7h  38ffiJ  E-  std- 

T. 13h 10m 

but  the  moon's  right  ascension  and  declina- 
tion are  changing  so  rapidly  that  we  must  reach  the  result  by  approx- 
imation. We  may  first  assume  the  moon's  place  at  rising  to  be 
the  same  as  at  standard  noon,  December  25  (or  5h,  G.M.T.),  and  at 
that  time  the  Almanac  gives  the  moon's  right  ascension  8h  54m, 
Decl.  H-  14°.9.  Marking  this  place  on  the  globe  and  bringing  it 
to  the  eastern  horizon,  we  find  E.A.M.  =  lh  56m,  and  continue  the 
computation  as  in  the  second  column  of  the  table  below.  (See 
Example  15.) 

G.M.T.  5h  12h  28m  12h  45™ 

Moon's  Place     8h  54™,  +  14°. 9         9h  llm,  +  13°.9         9h  13m,  +  13°.9 

R.A.M.      .     .     .       lh  56m  2h  13™  2h  18m 

R.A.  of  Sun.     .  18     12  18     12  18     12 

App.  T.      ...       7     44  8 

Eq.  of  T.   .     .     .  0_ 

Camb.  M.  T.       .       7     44 
Red.  for  Long.    .         —  16 


E.  Std.  T.  .  7     28  7     45  7     50 


132 


LABORATORY  ASTRONOMY 


FIG.  64.  Moonrise  at  Cam- 
bridge December  25, 1904 : 
K.A.M.  2*  18» 


This  gives  as  the  approximate  time  of  moonrise  7h  28m,  E.  Std.  T., 
or  12h  28m,  G.M.T.,  and  finding  the  moon's  place  for  this  time, 
R.A.  9h  llm,  Decl.  +  13°.9,  we  better  our  result  by  the  computation 
shown  in  the  third  column,  which  gives 
7h  45m,  E.  Std.  T.,  or  12h  45m,  G.M.T.  With 
this  value  we  find  the  moon's  place  9h  13m, 
•f  13°.9,  and  compute  as  in  the  last  column, 
finding  E.  Std.  T.  =  7h  50m. 

As  this  is  within  ten  minutes  of  the  time 
for  which  the  data  were  assumed,  and  since 
in  ten  minutes  the  moon's  right  ascension, 
as  shown  by  the  difference  column,  changes 
by  24s,  —  a  quantity  too  small  to  be  surely 
measured  on  an  ordinary  10-inch  globe,  — 
we  may  regard  the  last  solution  as  suffi- 
ciently accurate. 

It  would  appear  that  the  two  last  results 
should  be  in  closer  agreement,  since  the  difference  in  the  assumed 
times  is  only  seventeen  minutes ;  the  two  first  measures,  however, 
were  not  made  with  care,  as  only  approxi- 
mate values  were  sought. 

It  is  obviously  an  advantage  to  estimate 
the  approximate  time  of  moonrise  as  closely 
as  possible  before  beginning  the  solution : 
this  may  be  done  by  noting  the  age  of  the 
moon  (page  IV  of  the  month)  and  remem- 
bering that  the  moon  rises  and  sets  about 
48m,  or  Oh.8,  later  each  night  than  the  night 
before,  and  that  at  new  moon  sun  and  moon 
rise  and  set  together.  Assuming  that  the 
sun  rises  at  6  A.M.  and  sets  at  6  P.M.,  stand- 
ard time,  we  shall  find  an  approximate  value 
of  the  standard  time  of  moonrise  or  moonset 
by  adding  to  these  times  a  number  of  hours 
equal  to  eight-tenths  of  the  moon's  age  in  days.  Thus,  in  the  pre- 
ceding problem,  the  moon's  age  being  eighteen  days  on  December 
25,  we  add  0.8  x  18h  =  14h.4  to  6  A.M.  to  find  the  time  of  moonrise ; 


FIG.  65.  Moonset  at  Cam- 
bridge December  18, 1904 : 
R.A.M.  9*  41"° 


EXAMPLES  OF  THE   USE  OF  THE   GLOBE 


133 


this  gives  8h.4  P.M.  as  the  approximate  time,  which  is  within  an  hour 
of  the  final  result. 

Example  29.  Find  the  time  at  which  the  moon  sets  at  Cam- 
bridge December  18,  1904. 

The  moon's  age  is  found  by  the  Ephemeris  to  be  eleven  days  ; 
hence  we  add  9h  to  6h  P.M.,  and  have  as  the  approximate  time  of 
moonset  15h,  corresponding  to  20h,  G.M.T.  We  may  record  the 
successive  approximations  as  follows : 


Assumed  G.M.T. 
Moon's  R.A.  and  Decl. 


FIRST 
APPROXIMATION 

20h 


SECOND 
APPROXIMATION 


2h 


R.A.M.  at  moonset 
Sun's  R.A.        .     . 

App.  T 

Eq.  of  T.      .     .     . 
Red.  for  Long. 
Std.  T. 


17 


39m 
46 


15     53 
-    4 

-16 
15    33 


20h  33m 

Qh  41m 

17     46 
15     55 

-20 
15     35 


A  single  recomputation  will  always  be  sufficient  if  the  moon's 
place  is  first  determined  by  computing  from  its  age. 


MISCELLANEOUS  EXAMPLES 

Example  30.  Find  the  duration  of  twilight 
at  Cambridge  March  1. 

Evening  twilight  ends  when  the  sun  has 
sunk  so  far  below  the  horizon  that  his  direct 
rays  can  no  longer  fall  upon  and  be  reflected 
by  any  particles  in  that  portion  of  the  atmos- 
phere which  lies  above  the  plane  of  the  hori- 
zon. This  is  usually  assumed  to  be  the  case 
when  the  sun  is  18°  below  the  horizon. 

Bringing  the  sun's  place  for  March  1  to  the 
horizon,  and  then,  by  means  of  the  extension  of  the  altitude  arc,  to 
point  18°  below  the  horizon  (Fig.  66),  we  have  the  following  values 

R.A.M.  at  sunset     .     . 4h  20m 

R.A.M.  at  end  of  twilight 6 0_ 

Difference  .  1     40 


FIG.  66.    End  of  Twilight  at 
Cambridge  March  1 


134 


LABORATORY  ASTRONOMY 


which  equals  the  change  in  the  sun's  hour-angle,  or  the  time  elapsed 
between  sunset  and  the  end  of  twilight. 

Example  31.  At  what  hour,  apparent  time,  does  morning  twi- 
light begin  at  Cambridge  June  21? 

June  21.    Sim's  place  18°  below  E.  horizon,  R.  A. M 20h  8m 

Sun's  R.A.  by  globe    .     . .       6    0 

App.  T 14  .  8 

Or 28  A.M. 

Example  32.  At  what  point  of  the  horizon  does  the  first  glim- 
mer of  dawn  appear  in  latitude  42°.4  on  June  21? 

Bringing  the  sun's  place  by  trial  to  the  altitude  arc  at  a  point 
18°  below  the  horizon  (Fig.  67),  the  reading  on  the  horizon  at  the 
graduated  edge  of  the  altitude  arc  is  E.  57° 
N.  =  Az.  213° ;  and  as  this  is  the  nearest 
point  of  the  horizon  to  the  sun  when  it  is 
18°  below  the  horizon,  it  is  at  this  point  or 
a  little  to  the  south  that  the  first  light  will 
appear. 

Example  S3.  How  many  hours  can  the 
sun  shine  into  north  windows  June  21  in 
latitude  41°? 

By  the  method  of  Example  15,  it  is  found 
that  the  apparent  times  of  sunrise  and  sun- 
set on  June  21  are  4h  30m  A.M.  and  7h  30m 
FIG.  67.  Dawn  at  Cambridge  P.M.,  and  by  the  method  of  Example  20, 
•june2i,at2fr8»A.M.:  Sun's  that  the  sun  is  in  the  prime  vertical  at  7h 

Az    213° 

56m  A.M.  and  4h  4m  P.M.  Hence  from  4h  30m 

to  7h  56m  A.M.  and  from  4h  4m  to  7h  30m  P.M.,  a  total  of  6h  52m,  the 
sun  shines  on  the  north  face  of  an  east  and  west  wall.  The  length 
of  the  day  is  fifteen  hours. 

Example  84.  August  20,  in  latitude  42£°,  longitude  4h  48m,  at 
ten  minutes  past  10  A.M.,  Eastern  standard  time,  the  sun  begins  to 
shine  upon  the  front  wall  of  a  building.  How  does  the  building  face  ? 

Since  at  the  given  time  the  sun  is  in  the  same  vertical  plane 
with  the  front  wall  of  the  building,  the  problem  requires  us  to 
determine  the  direction  of  this  plane  by  finding  the  sun's  azimuth, 
which  may  be  done  as  follows : 


EXAMPLES  OF  THE  USE  OF  THE   GLOBE  135 

Rectifying  for  latitude  42^-°,  we  have : 

Std.  T.  10h  10m  A.M =  22h  10m 

Red.  for  Long,  (from  E.  Std.  T.) +  12 

Local  M.T 22    22 

Subtract  Eq.  of  T.  (additive  to  App.  T.)     .     .     .     . '  .     .  -  3 

App.  T 22     19 

Sun's  R.A 10 1_ 

Sid.  T 8    20 

Eectifying  for  this  time  and  bringing  the  altitude  arc  to  the 
sun's  place  for  August  20,  we  find  the  sun's  azimuth  to  be  315°. 
Hence  the  front  wall  is  in  a  line  from  southeast  to  northwest,  and 
the  building  fronts  southwest. 

Example  35.  What  is  the  greatest  north- 
ern latitude  in  which  all  of  the  four  bright 
stars  of  the  Southern  Cross  are  visible? 
What  must  be  the  time  of  year  ? 

Rectifying  the  globe  for  the  equator,  the 
Southern  Cross  (about  R.A.  12h,  Decl.  -  60°) 
is  brought  to  the  meridian  and  the  brass 
meridian  is  moved  in  its  own  plane  until 
the  lowest  star  is  brought  to  the  horizon  at 
its  south  point.  The  elevation  of  the  pole 
above  the  north  horizon  is  then  read  on  the 
brass  meridian  and  found  to  be  28°,  which  FIG.  68.  August  20:  std.T.io* 
is  the  required  latitude.  The  star  being  still  10m;  R;A-M-  8h  ""J  Sun'8 

Az.  315 

in  the  same  position,  the  altitude  arc  is  then 

used  to  mark  the  points  of  the  ecliptic  which  are  18°  below  the 
horizon.  These  are  found  to  be  at  points  occupied  by  the  sun 
January  2  and  May  25,  and  between  these  dates,  therefore,  the 
whole  cross  may  be  above  the  horizon  in  latitude  28°  in  the  full 
darkness  of  night,  the  sun  being  below  the  twilight  limit. 

Example  36.  What  is  the  latest  date  at  which  we  can  see  Sirius 
in  the  evening  twilight  in  latitude  42°? 

Sirius  is  visible  when  the  sun  is  about  10°  below  the  horizon,  and 
cannot  be  seen  later  than  the  day  on  which  he  sets  at  the  instant 
that  the  sun  is  10°  below  the  horizon. 

Rectifying  for  42°  and  bringing  Sirius  to  the  western  horizon,  we 
find  that  the  point  of  the  ecliptic  which  is  10°  below  the  horizon  is 


136 


LABORATORY   ASTRONOMY 


the  place  occupied  by  the  sun  on  May  15,  which  is,  therefore,  the 
required  date. 

Example  37.  Between  what  dates  is  the  sun  visible  at  midnight 
at  the  North  Cape,  in  latitude  70°  north  ? 

Eectifying  the  globe  for  70°  north  and  rotating  the  globe  slowly, 
it  is  found  that  points  on  the  ecliptic  in  longitudes  58°  and  122°  can 
be  brought  exactly  to  the  north  point  of  the  horizon ;  any  point 
between  these  may  be  brought  to  the  meridian  below  the  pole  and 
above  the  horizon.  The  dates  at  which  the  sun  occupies  these  posi- 
tions are  May  19  and  July  25,  and  between  these  dates  the  sun  will 
always  come  to  the  meridian  at  midnight  above  the  horizon. 

Example  38.  Illustrate  the  "  harvest  moon  "  by  finding  the  time 
of  moonrise  at  Edinburgh,  latitude  56°,  on  successive  dates  about 
the  time  of  full  moon,  September  24,  1904. 

As  only  approximate  results  are  desired,  we  may  take  from  the 
Ephemeris  the  moon's  place  for  6h  P.M.,  G.M.T.,  and  solve  as  follows: 


1904 

R.A. 

DECL. 

K.A.M.  AT 
MOONRISE 

SUN'S  ll.A. 

APPARENT 
TIME 

September  22 

22h  36m 

-8° 

17h  22m 

12h     Om 

5h  22m 

23 

23     22 

-4 

17     42 

12      4 

5    38 

24 

0       7 

-  1 

18      9 

12       7 

0      2 

25 

0     52 

+  3 

18     28 

12     10 

6     18 

26 

1     38 

+  7 

18     51 

12    14 

6     37 

And  it  appears  that  the  moon  rises  about  twenty  minutes  latter 
each  night  than  it  did  on  the  previous  night. 

Example  39.  Find  the  time  of  moonrise  at  Edinburgh  on  succes- 
sive nights  at  full  moon,  March  31,  1904. 

We  have,  as  in  Example  38,  the  moon's  place  at  6h  P.M.,  G-.M.T. : 


1904 

R.A. 

DECL. 

R.A.M.  AT 

MOONRISE 

SUN'S  E.A. 

APPARENT 

TIME 

March  30    .    . 

11*  56m 

+  1Q 

5h    57m 

0^  38™ 

5h    IQm 

31    .   . 

12     52 

-4 

7      20 

0    41 

6     39 

April  1    .    .    . 

13    49 

-8 

8     42 

0    44 

7    58 

EXAMPLES  OF  THE  USE  OF  THE  GLOBE  137 

Therefore  the  full  moon  at  the  time  of  the  vernal  equinox  rises 
about  one  hour  and  twenty  minutes  later  each  night.  (Notice  and 
explain  the  difference  in  the  accuracy  attained  in  these  two 
examples.) 

Example  40-  Find  the  rate  at  which  8  Orionis  is  changing  its 
azimuth  at  rising  and  setting  in  latitude  42°. 

Rectifying  for  42°  and  bringing  8  Orionis  to  the  eastern  horizon, 
we  find  R.A.M.  =  23h  23m  ;  Az.  =  271°.  Increasing  the  hour-angle 
half  an  hour  by  making  R.A.M.  =  23h  53m,  we  find,  by  the  alti- 
tude arc,  Az.  =  276°.  Bringing  the  star  to  the  western  horizon,  we 
have  K.A.M.  =  llh  24m ;  Az.  =  89^°.  Decreasing  the  hour-angle 
by  making  E.A.M.  =  10b  54m,  we  find  Az.  =  84£°  half  an  hour 
before  setting.  In  both  cases  the  diurnal  rotation  causes  the 
azimuth  to  increase  at  the  rate  of  5°  in  half  an  hour. 

By  solving  the  same  problem  for  stars  in  various  parts  of  the 
heavens,  as,  for  instance,  Vega,  y  Pegasi,  Airfares,  and  a  Gruis,  it 
appears  that  stars  of  whatever  declinations,  when  near  the  horizon, 
are  increasing  their  azimuths  by  about  10°  per  hour  in  latitude  42°. 
(This  is  the  rate  at  which  the  plane  of  the  pendulum  appears  to 
revolve  in  Foucault's  experiment.) 

Example  41.  To  mark  the  hour-lines  on  a  horizontal  sundial  for 
use  in  latitude  42°. 

The  gnomon  of  an  ordinary  sundial  (Fig.  69)  is  directed  toward  the 
pole,  and  its  shadow  at  apparent  noon  falls  upon  the  horizontal  dial 
on  the  line  of  XII  hours,  which,  when 
properly  adjusted,  lies  in  the  direc- 
tion of  the  meridian.  The  shadow 
at  that  time  is  in  a  line  drawn  through 
the  foot  of  the  gnomon  toward  azi- 
muth 180°.  It  always  passes  through 
the  intersection  of  the  gnomon  with 
the  dial  and,  continually  shifting  F™-69-  Horizontal  sundial, 

/  Latitude  42° 

toward  the  east,  at  any  instant  lies 

in  the  plane  containing  the  sun  and  the  gnomon.  This  plane  cuts 
the  celestial  sphere  in  the  sun's  hour-circle.  The  shadow,  therefore, 
is  a  line  which  passes  through  the  foot  of  the  gnomon  and  whose 
azimuth  is  that  of  the  intersection  of  the  sun's  hour-circle  with 


138 


LABORATORY  ASTRONOMY 


the  plane  of  the  horizon.  For  a  given  hour-angle  the  position  of 
this  line  will  be  the  same  whatever  the  position  of  the  sun  upon 
its  circle,  and  is  therefore  the  same  for  a  given  apparent  time 
whatever  the  time  of  year. 

We  may  find  the  azimuth  of  the  intersection  of  a  given  hour- 
circle  with  the  horizon  by  means  of  the  globe  as  follows.  Rectify- 
ing the  globe  for  42°,  the  vernal  equinox  is  brought  to  the  meridian, 
so  that  the  equinoctial  colure  cuts  the  horizon  at  azimuth  180°. 
In  this  position  R.A.M.  is  Oh,  and  the  azimuth  of  the  shadow  is 
180°.  Increasing  the  hour-angle  of  the  colure  by  successive  incre- 
ments of  15°,  we  have  the  following  values  for  the  azimuths  of 
the  hour-lines  : 

FOB  THE  P.M.  HOURS: 


I 

II 
III 

IV 

V 

VI 

VII 


R.A.M. 

1* 

2 
3 

4 
5 
6 

7 


)URS: 

AND  SIMILARLY 
FOR  THE  A.M.  HOURS  : 

Azimuth  of 
Shadow 

R.A.M. 

Azimuth  of 
Shadow 

190 

XI 

23 

170 

201 

X 

22 

159 

214 

IX 

21 

146 

230 

VIII 

20 

130 

249 

VII 

19 

111 

270 

VI 

18 

90 

291 

V 

17 

69 

If  the  hour-circles  are  shown  for  each  15°  as  on  most  modern 
globes,  it  is  sufficient  to  bring  one  hour-circle  to  the  meridian  and 
note  the  points  where  the  other  circles  cut  the  horizontal  plane ; 
Fig.  57  shows  the  globe  rectified  to  42°  and  Oh  Sid.  T.,  and  therefore 
in  position  for  reading  the  azimuths  of  the  successive  hour-lines 
directly  on  the  horizon. 

Example  42.  To  mark  the  hour-lines  of  a  vertical  sundial  for  use 
in  latitude  42°  N.,  the  bearing  of  the  plane  being  W.  24°  S. 

Here  the  shadow  of  the  gnomon  falls  upon  a  vertical  plane, 
and  the  line  for  noon  is  a  vertical  line  through  the  intersection 
of  the  gnomon  with  the  plane. 

At  any  given  hour  after  noon  the  shadow  falls  below  the  gnomon 
and  to  the  east  of  the  XII  line  (Fig.  70),  since  it  marks  the  inter- 
section of  the  plane  of  the  dial  by  the  sun's  hour-circle.  It  makes 
an  angle  with  the  XII  line  which  may  be  defined  as  the  "  nadir 


EXAMPLES  OF  THE  USE  OF  THE  GLOBE 


139 


distance  "  of  the  line  of  intersection  of  the  two  planes,  and  this  is 
equal  to  the  zenith  distance  of  that  part  of  the  same  line  which 
lies  above  the  gnomon. 

This  problem  therefore  requires  us  to  find 
the  zenith  distance  of  the  intersection  of  the 
sun's  hour-circle  with  the  vertical  plane  for  a 
given  hour-angle  of  the  sun,  and  may  be  solved 
with  the  globe  as  follows  : 

Rectify  the  globe  for  latitude  42°,  and  adjust 
the  altitude  arc  to  the  zenith  with  its  foot  at 
azimuth  66°  on  the  horizon ;  its  plane  then 
corresponds  to  that  of  the  dial. 

Bringing  the  vernal  equinox  to  the  meridian,  R.A.M.  =  Oh,  the 
equinoctial  colure  intersects  the  altitude  arc  at  zenith  distance  0°. 
Increasing  the  hour-angle  of  the  colure,  as  in  Example  41,  we  have 
successively 


FIG.  70.  Vertical  Dial, 
Latitude  42° 


HOUK-LlNE 

I 

II 
III 
IV 

V 


R.A.M. 

lh 

2 
3 
4 
5 


ZENITH  DISTANCE 
OF  INTERSECTION 

13° 
30 
49 
70 
•        90 


which  gives  the  angles  of  the  afternoon  lines  from  the  noon  line. 
Setting  the  arc  at  azimuth  246°,  we  find  in  the  same  way 


HOUR-LINE 

XI 

X 

IX 

VIII 

VII 

VI 

V 


R.A.M. 

23* 

22 

21 

20 

19 

18 

17 


ZENITH  DISTANCE 
OF  INTERSECTION 

11° 

22 

33 

44 

56 

70 

90 


which  gives  the  morning  lines.  The  A.M.  and  P.M.  divisions  will 
not  be  symmetrical  about  the  XII  line  unless  the  vertical  plane 
faces  due  south. 

Example  4&-  Find  the  path  of  the  shadow  of  a  pin  head  on  a 
horizontal  plane  at  Cambridge  March  21,  from  8  A.M.,  apparent 
time,  to  5  P.M.,  apparent  time. 


140 


LABORATORY  ASTRONOMY 


Rectifying  the  globe  for  latitude  42°,  bringing  the  sun's  place  to 
hour-angles  which  correspond  to  the  successive  hours  from  8  A.M. 
to  5  P.M.,  and  measuring  its  altitude  and  azimuth  in  each  position 
by  the  altitude  arc,  we  have  the  following  results  : 


APP.  TIME 

RA.M. 

ALTITUDE 

AZIMUTH 

DISTANCE 

8h  A.M. 

20h 

22° 

291° 

12.5cm. 

9 

21 

32 

303 

8.1 

10 

22 

40 

318 

6.0 

11 

23 

46 

337 

4.9 

Noon 

0 

48 

0 

4.5 

lh  P.M. 

1 

46 

22 

4.9 

2 

2 

40 

42 

6.0 

3 

3 

32 

56 

8.0 

4 

4 

22 

C8 

12.6 

5 

5 

11 

80 

26.5 

To  construct  the  curve  we  must  know  the  length  of  the  pin ; 
assuming  this  to  be  5  cm.  long,  a  point  on  the  paper  is  chosen  to 
represent  the  point  vertically  under  the  pin  head,  and  through  it 
is  drawn  a  line  to  represent  the  meridian,  and  other  lines  are 


FIG.  71.    Azimuth,  of  Shadow 

drawn  at  the  azimuths  differing  by  180°  from  those  given  in  the 
above  table.  (See  Fig.  71.)  The  shadow  path  will  cross  these 
lines  at  the  corresponding  hours. 

To  find  the  distance  of  any  point  of  the  shadow  path  from  the 
foot  of  the  pin,  we  may  reverse  the  process  explained  on  page  5. 
Drawing  a  line  from  C,  the  center  of  the  base  in  Fig.  6,  through 
the  divisions  of  the  protractor  corresponding  to  any  one  of  the  alti- 
tudes of  the  above  table  and  measuring  the  line  A'B',  we  have  the 
distance  in  centimeters  from  the  foot  of  the  pin  to  the  point  where  the 
shadow  falls  on  the  corresponding  azimuth  line.  The  last  column 
of  the  above  table  gives  the  distances  measured  in  this  manner. 


EXAMPLES  OF  THE  USE   OF  THE   GLOBE 

Fig.  72  shows  the  shadow  path  as  thus  constructed,  and  it  is 
evidently  a  straight  line.  This  will  always  be  the  case  on  the  day 
of  the  equinox,  when  the  sun  is  in  the  equator  and  its  diurnal  path 
is  consequently  a  great  circle. 


FIG.  72.    Path  of  Shadow 


THE   HOUR-INDEX 

The  globe  is  usually  provided  with  an  arrangement  by  means  of 
which  approximate  solutions  may  be  made  of  problems  involving 
time  without  the  use  of  the  graduations  of  the  equinoctial. 

This  process  is  so  simple  that  its  explanation  might  well  have 
preceded  that  of  the  method  of  finding  the  sun's  hour-angle  given 
on  page  112  and  used  in  Problem  7.  It  is,  however,  very  inaccurate, 
and  should  only  be  chosen  where  an  error  of  several  minutes  is 
unimportant. 

The  most  convenient  form  given  to  the  attachment  is  that  of  a 
small  pointer  fixed  to  the  brass  meridian  in  such  a  manner  that  it 
revolves  about  the  same  center  as  the  polar  axis,  but  with  sufficient 
friction  to  keep  it  fixed  in  any  •position  where  it  may  be  placed. 

This  pointer,  or  "  hour-index,"  lies  close  to  the  surface  of  the 
globe,  which  revolves  freely  under  it.  The  end  of  the  index  lies 
over  a  small  circle  on  the  globe,  about  15°  from  the  pole ;  and  this 
circle  is  graduated  into  hours  and  quarters  in  two  groups  of  12 
hours  each,  numbered  in  the  same  direction  as  the  graduations  of 
the  equinoctial. 

The  following  example  illustrates  the  use  of  the  hour-index, 
which  in  this  case  gives  sufficiently  good  results  with  less  trouble 
than  the  method  already  explained. 

Example  44.  Find  the  apparent  times,  October  1,  1904,  of  rising 
and  setting  of  the  planets  whose  places  are  given  on  page  130. 


142  LABORATORY  ASTRONOMY 

Mark  the  places  of  the  planets  and  of  the  sun  ;  bring  the  latter  to 
the  meridian  and  set  the  hour-index  to  read  XII  noon.  Eotate  the 
globe  through  any  angle,  and  the  reading  of  the  index  will  equal 
the  hour-angle  of  the  sun  in  its  new  position,  and  thus  will  give 
directly  the  corresponding  apparent  time. 

We  may,  therefore,  rapidly  determine  the  apparent  time  of  rising 
and  setting  of  all  the  planets  by  bringing  each  in  turn  to  the  eastern 
and  western  horizon  and  noting  the  reading  of  the  hour-index. 

The  hour-index  may  be  adjusted  to  give  local  mean  time  or 
standard  time  directly  by  making  it  read  the  local  mean  time  or 
standard  time  of  apparent  noon  when  the  sun  is  brought  to  the 
meridian.  Thus,  for  October  1,  at  Cambridge,  longitude  4h  44m  : 

App.  T.  of  App.  noon 12h    Om 

Eq.  of  T -10 

Camb.  M.  T.  of  App.  noon 11     50 

Red.  for  Long —  16 

Std.  T.  of  App.  noon 11     34 

And  the  index  should  be  set  to  read  llh  34m  when  the  sun  is  on 
the  meridian,  in  order  to  give  Eastern  standard  time. 


CHAPTER  X 
THE   MOTIONS   OF   THE   PLANETS 

IT  has  been  the  aim  of  the  preceding  chapters  to  show  how  the 
diurnal  motion  and  the  motion  of  the  sun  and  moon  among  the 
stars  may  be  studied  in  such  a  manner  that  the  student  shall  acquire 
and  fix  his  knowledge  in  large  part  by  his  own  observations. 

There  remains  to  be  considered  the  motion  of  the  planets,  which 
cannot  be  studied  in  the  same  way  because  they  move  so  slowly 
that  a  long  time  would  be  required  to  obtain  a  sumcient  number 
of  observations  on  which  to  base  a  satisfactory  theory.  It  is  of 
course  desirable,  however,  during  the  continuance  of  the  observa- 
tions on  the  moon  and  stars  to  include  the  planets  in  order  to 
establish  a  few  fundamental  facts,  such  as  that  they  never  appear 
far  from  the  ecliptic  and  that  in  general  they  move  from  west  to 
east  like  the  sun  and  moon,  but  that  when  opposite  the  sun,  so 
that  they  come  to  the  meridian  at  midnight,  they  are  moving  from 
east  to  west  among  the  stars.  Their  places  in  the  heavens  should 
be  occasionally  observed,  for  comparison  with  the  places  derived 
from  the  theory  which  forms  the  subject  of  the  present  chapter. 

In  treating  of  this  theory  we  shall  first  assemble  the  few  prin- 
ciples which  have  been  shown  to  account  for  the  observed  motions, 
and  shall  then  show  how  these  principles  may  be  applied  to  the 
graphical  solution  of  problems  involving  the  determination  of  the 
place  in  the  heavens  of  a  planet  as  seen  from  the  earth  at  any 
given  time.  These  problems  serve  to  illustrate  and  explain  the 
phenomena  resulting  from  the  planetary  motions,  as  the  globe 
problems  of  the  preceding  chapter  serve  for  those  resulting  from 
the  diurnal  rotation  of  the  earth. 

Results  of  the  Law  of  Gravitation.  —  In  consequence  of  the  attrac- 
tion of  the  sun,  each  planet  describes  an  ellipse,  having  the  sun  in 
one  focus  ;  this  is  "Kepler's  first  law."  The  mutual  attractions  of 
the  planets  produce  "  perturbations  "  of  their  motion,  but  in  no  case 

143 


144 


LABORATORY  ASTRONOMY 


are  these  perturbations  sufficient  to  alter  the  place  of  the  planet 
by  so  much  as  one  degree  from  its  place  as  determined  by  the  sun's 
attraction.  Jupiter  may  be  displaced  about  0°.3  and  Saturn  nearly 
0°.8 ;  but  with  this  exception  no  displacement  of  a  planet  amounts 
to  J°.  The  asteroids  are  subject  to  much  greater  perturbations. 

The  orbit  of  each  planet  is  in  a  plane  which  remains  nearly  fixed, 
and  the  planes  of  all  the  orbits  are  so  nearly  coincident  with  the 
ecliptic  that  the  projections  of  their  paths  on  the  ecliptic  are  no 
more  distorted  than  the  roads  of  a  moderately  rugged  country  are 
distorted  in  their  representations  on  an  ordinary  plane  map.  This 
fact  makes  it  as  easy  to  determine  their  motions  by  an  accurate 
map  of  their  orbits  on  the  plane  of  the  ecliptic  as  to  follow  the 
motion  of  a  traveler  over  a  well-charted  country,  when  his  point 
of  departure  and  rate  of  travel  are  known. 


PROPERTIES  OF  THE  ELLIPSE 

An  ellipse  may  be  drawn  by  putting  two  pins  upright  in  a  board, 
as  in  Fig.  73,  laying  a  knotted  loop  of  thread  on  the  board  so  as  to 
include  both  pins,  and  then  putting  the  point  of  a  well-sharpened 
pencil  on  the  surface  inside  the  loop.  Let  the  pencil  be  moved  out 


FIG.  73.    Drawing  an  Ellipse 

so  as  to  form  the  loop  into  a  triangle,  and  then  drawn  along  the 
surface  so  as  to  pass  successively  through  all  the  points  which  it 
can  reach  without  allowing  the  thread  to  become  slack.  The  curve 
which  it  follows  will  be  an  ellipse  whose  shape  and  size  will  depend 
only  on  the  distance  between  the  pins  and  the  size  of  the  loop. 

The  form  of  the  curve  is  shown  in  Fig.  74. 

F1  and  F2  are  the  foci,  AB  the  major  axis,  and  C,  which  bisects 
both  F^FZ  and  AB,  is  the  center  of  the  ellipse.  PF-^  is  the  radius 


THE  MOTIONS  OF  THE  PLANETS  145 

vector  from  any  point  P  to  Flf  and  PF2  the  radius  vector  to  F2. 
They  are  usually  represented  by  r±  and  r2.  r±  +  rz  is  a  constant  for 
all  points  of  the  ellipse,  being  always  equal  to  the  length  of  the 
thread  minus  FiFz.  For 
the  point  A 


and  since  from  the  sym- 
metry of  the  curve  2? 

AFl  =  BF2, 


A  C  is  usually  represented 
by  a,  and  CF-,  or  CF2  by  c. 

J  FIG.  74.    Fundamental  Points  and  Lines 

Since    2  c   equals    the 

distance  between  the  foci,  and  2  a  +  2  c  the  length  of  the  thread, 
the  shape  and  size  of  the  ellipse  are  completely  fixed  by  the  values 
of  a  and  c.  The  ratio  c/a  is  called  the  eccentricity  and  is  repre- 
sented by  e  ;  it  is  always  less  than  unity.  The  line  along  which 
the  major  axis  lies  is  called  the  line  of  apsides. 

To  draw  a  Given  Ellipse.  —  Let  it  be  required  to  draw  an  ellipse 
whose  semi-major  axis  is  one  inch,  and  eccentricity  i,  with  one 
focus  at  the  point  F1  of  Fig.  75,  and  with  its  major  axis  inclined 
30°  to  the  horizontal. 

Draw  the  line  of  apsides  MN  at  the  proper  angle.  Since  e  =  J,  we 
locate  C  one-fourth  of  an  inch  from  Fl  on  the  line  of  apsides. 
Take  F2  at  an  equal  distance  beyond  C,  make  the  total  length  of 
the  thread  2£  inches  =  2  a  -f-  2  c,  and  draw  the  ellipse  as  shown  in 
the  figure. 

The  dotted  line  surrounding  the  ellipse  is  a  circle  drawn  about 
C  as  a  center  with  a  radius  of  one  inch  (equal  to  the  semi  major 
axis).  It  is  worthy  of  notice  that  the  ellipse  differs  but  little 
from  this  circle,  the  greatest  distance  between  the  two  being 
about  y^  of  an  inch.  With  a  less  eccentricity  the  agreement  of 
the  two  curves  is  closer.  For  e  =  0.10  the  difference  is  but  .005 
of  the  semi  major  axis,  so  that  an  ellipse  of  that  eccentricity  whose 
semi  major  axis  is  two  inches  differs  at  no  point  more  than  T£^  of 


146 


LABORATORY  ASTRONOMY 


an  inch  from  a  circle  struck  about  its  center  with  a  radius  of  two 
inches.  If  the  orbits  of  the  planets  are  drawn  with  their  true 
eccentricities  and  with  a  line  0.01  inch  in  width,  and  in  each  case 
a  circle  is  struck  with  radius  a  about  the  center  of  the  ellipse,  and 
having  a  width  of  .01  inch,  no  white  space  will  be  anywhere  visi- 
ble between  the  two  lines  unless  the  diameter  of  the  circle  is  greater 


Horizontal 


FIG.  75.    Ellipse  drawn  with  Given  Constants 

than  about  1  inch  for  Mercury,  4^  inches  for  Mars,  17  inches  for 
Jupiter,  and  12^  inches  for  Saturn.  For  Venus  and  the  earth  the 
circles  may  be  several  feet  in  diameter.  The  orbits  may  therefore 
be  represented  by  such  circles  with  a  considerable  degree  of 
accuracy. 


MEAN   AND   TRUE   PLACE   OF   A   PLANET 

Having  considered  the  geometrical  properties  of  the  planetary 
orbits,  it  is  next  in  order  to  inquire  as  to  the  law  which  regulates 
the  motions  of  the  planets  in  their  orbits. 

Since  the  sun  is  at  one  focus  of  the  orbit,  the  planet's  distance 
from  the  sun  varies  continually.  It  is  nearest  the  sun  at  the  peri- 
helion point,  which  is  at  one  extremity  of  the  major  axis.  Aphelion 
occurs  at  the  opposite  end  of  the  major  axis,  and  the  planet  is  then 
at  its  greatest  distance. 


THE  MOTIONS  OF  THE  PLANETS 


147 


Kepler's  second  law  states  that  the  planet  moves  in  such  a  way  that 
its  radius  vector  sweeps  over  equal  areas  in  equal  times.  The  appli- 
cation of  this  principle  will  be  evident  from  the  following  illustration. 

Fig.  76  represents  the  orbit  of  Mercury  in  its  true  proportions. 
The  period  of  the  revolution  of  the  planet  is  eighty-eight  days,  in 
which  time  the  radius  vector  sweeps  over  the  whole  area  of  the 
ellipse.  To  pass  from  perihelion  to  aphelion  would  require  forty- 
four  days,  or  one-half  the  period,  since  the  area  described  is  one- 
half  the  area  of  the  whole  ellipse.  It  is  not  difficult  to  fix  very 
nearly  the  point  reached  by  the  planet  twenty-two  days  after  pass- 
ing through  perihelion.  It  will  then  have  accomplished  a  quarter 
of  a  revolution,  and  be  at 
such  a  point  P  that  the  area 
ASP  is  one-quarter  of  the 
ellipse,  or  one-half  of  A  PBS, 
so  that  APS  equals  BPS. 

It  may  be  shown  that  this 
point  must  be  very  nearly  in 
the  line  Pf  drawn  perpen- 
dicular to  the  major  axis 
through  f,  the  "  empty " 
focus  of  the  orbit,  as  it  is 
sometimes  called. 

Assuming  P  to  be  on  this 
line,  and  drawing  a  perpen- 
dicular Sk  through  the  focus 
occupied  by  the  sun,  and  also  the  radius  vector  PS,  we  have  from 
the  symmetry  of  the  ellipse,  Area  ASk  equal  Area  BfP,  and  the 
triangle  PkS  evidently  equals  the  triangle  PfS.  The  difference  of 
the  two  areas  ASP  and  BSP  is  therefore  the  segment  of  the  ellipse 
cut  off  by  the  chord  Pk  ;  this  segment  is  so  very  small  that  the 
area  ASP  is  very  nearly  equal  to  BSP. 

The  angle  ASP  through  which  the  planet  has  moved  about  the 
sun  since  perihelion  is  called  its  "  true  anomaly."  In  this  case  it 
is  about  110°.  We  may  now  infer  that  the  true  anomalies  of  Mer- 
cury 22,  44,  66,  and  88  days  after  perihelion  would  be  about  110°, 
180°,  250°,  and  0°,  respectively. 


FIG.  76.   Equal  Areas  in  the  Ellipse 


148 


LABORATORY  ASTRONOMY 


It  is  convenient  to  refer  the  motion  of  the  planet  to  that  of  a 
hypothetical  planet  moving  in  the  orbit  in  such  a  way  as  to  be  at 
perihelion  with  the  real  planet  and  describe  equal  angles  in  equal 
times ;  thus  the  anomaly  of  the  so-called  "  mean  planet "  after  22, 
44,  66,  and  88  days  would  be  90°,  180°,  270°,  and  360°,  respectively. 
The  Equation  of  Center.  —  The  quantity  to  be  added  to  the  anomaly 
of  the  mean  planet,  or  briefly,  the  "  mean  anomaly  "  of  the  planet,  in 
order  to  find  its  true  anomaly,  is  called  the  "equation  of  center"; 
in  the  cases  above  given  it  is  for  the  four  positions  0°,  +  20°,  0°,  and 
—  20°.  It  is  always  positive  for  values  of  the  mean  anomaly  between 

0°  and  180°,  and  negative 
for  values  between  180° 
and  360°.  It  appears  from 
Fig.  77,  in  which  P  and  P' 
mark  the  true  and  mean 
places  of  the  planet  re- 
spectively, that  at  all 
points  from  perihelion  A 
to  aphelion  B,  the  true 
anomaly  ASP  is  greater 
than  the  mean  anomaly 
A  SP',  while  from  aphelion 
to  perihelion  ASP  is  less 
than  ASP'. 

The  value  of  the  mean 
anomaly  being  given  for 
any  time,  its  value  for  any  other  time  is  easily  found,  since  it 
increases  uniformly  from  0°  to  360°  in  the  time  required  for  the 
planet  to  make  one  revolution. 

The  mean  anomaly  being  known,  we  may  pass  to  the  true  anomaly 
by  means  of  a  table  of  the  equation  of  center  (page  174),  in  which 
the  value  of  the  latter  is  given  for  each  degree  or  ten  degrees  of 
the  planet's  mean  anomaly. 

The  computation  of  these  tables  lies  far  beyond  our  scope,  but  it 
is  worth  while  to  note  that  approximate  values  of  the  equation  of 
center  may  be  found  by  a  graphical  method,  which  rests  upon  the 
principle  that  in  describing  equal  areas  about  one  focus  of  an 


FIG.  77 


THE   MOTIONS  OF  THE  PLANETS 


149 


ellipse  of  small  eccentricity,  a  planet  describes  very  nearly  equal 
angles  about  the  other  focus. 

If  then  the  ellipse  be  carefully  constructed  on  a  large  scale,  say 
with  a  major  axis  of  ten  inches,  and  through  the  empty  focus  lines 
be  drawn  making  angles  of  10°,  20°,  30°,  etc.,  with  the  line  of 


F2 


FIG.  78 


apsides,  these  lines  will  cut  the  ellipse  at  the  places  occupied 
by  the  true  planet  when  its  mean  anomalies  are  10°,  20°,  30°,  etc. 
Fig.  78  shows  one-half  of  the  orbit  of  Mercury  divided  into  six 
equal  parts  in  this  manner. 

The  true  places  being  thus  fixed,  and  lines  drawn  from  each  to 
the  sun,  the  true  anomalies  may  be  read  off  with  a  protractor  ;  and 
by  comparison  with  the  mean  anomalies  the  equation  of  center  for 
each  ten  degrees  of  mean  anomaly  may  be  determined. 


MEASUREMENT  OF  ANGLES  IN  RADIANS 

It  has  been  assumed  that  the  student  is  familiar  with  the  ordi- 
nary method  of  measuring  angles  in  degrees.  For  some  purposes 
it  is  convenient  to  select  a  different  unit,  the  u  radian." 

One  radian  is  the  angle  subtended  by  an  arc  whose  length 
(measured  by  a  flexible  scale  laid  along  the  curve  of  the  arc)  is 
equal  to  that  of  the  radius.  This  angle  measured  in  the  ordinary 
way  is  found  to  be  57°.3  =  3438',  or  206,265". 

If  the  length  of  an  arc  a  is  known,  and  also  the  radius  of  the 
circle  r,  the  angle  subtended  by  the  arc  is  a/r  (arc  -^  radius)  radians. 
Thus  in  a  circle  two  feet  in  diameter,  an  arc  of  one  inch  subtends 
an  angle  of  1/12  radian,  —  6  inches  of  0.5  radian,  1  foot  of  1  radian, 
etc.  Since  1  radian  equals  57°. 3,  an  arc  of  one  inch  in  the  above  circle 


150  LABORATORY  ASTRONOMY 

subtends  1/12  x  57.3°;  and,  in  general,  radians  are  transformed  to 
degrees,  minutes,  or  seconds  of  arc  by  multiplying  by  57.3,  3438, 
and  206,265,  respectively;  and  degrees,  minutes,  or  seconds  to 
radians  by  dividing  by  57.3,  3438,  and  206,265,  respectively. 

The  use  of  the  radian  is  especially  convenient  in  problems  in- 
volving an  angle  so  small  that  the  corresponding  arc  nearly  equals 
its  chord  or  the  perpendicular  drawn  from  one  extremity  of  the  arc 
to  the  radius  drawn  through  its  other  extremity.  The  method  is 
illustrated  by  the  following  instances  : 

1.  The  moon's  distance  is  240,000  miles,  and  its  angular  diameter 
is  31',  or  31/3438  radian.  Its  diameter  in  miles  is  given  by  the 
equation 

24^00  =  3§8'     HenC6  D  =  2164  miles'  approximately. 


2.  The  height  of  a  tree  is  30  feet,  and  the  length  of  its  shadow 
is  150  feet.    The  altitude  of  the  sun  is 

a/r  =  30/150  =  0.2  radian  =  11°.46. 

The  true  value  obtained  by  trigonometrical  computation  is  11°.54, 
differing  by  .08°,  and  this  approximate  method  will  give  results 
within  0°.l  so  long  as  the  angle  does  not  exceed  this  value. 

3.  By  means  of  a  sextant  the  angle  between  the  water  line  of  a 
distant  war  ship  (Fig.  79)  and  the  top  of  its  military  mast  is  found 


.O-OiL- 


FIG.  79 


to  be  17'  10".  The  height  of  the  mast  is  known  to  be  120  feet. 
Assuming  this  height  to  be  equal  to  the  arc  subtended  by  the 
measured  angle,  we  have 

17'  10"  =  0.005  radian  =  ?  =   beight  of  °">Bt 

r       distance  of  ship 

and  the  distance  of  the  ship  is  about  8000  yards. 


THE  MOTIONS  OF  THE  PLANETS  151 

DIAGRAM  OF  CURTATE   ORBITS 

Fig.  80  represents  a  diagram  of  the  orbits  of  the  five  inner 
planets  projected  on  the  plane  of  the  ecliptic,  which  serves  to  solve 
many  problems  regarding  the  planetary  motions.  The  diagram  is 
of  convenient  size  for  actual  use,  if  its  dimensions  are  such  that 
one  astronomical  unit  equals  about  f  of  an  inch. 

In  order  to  show  how  small  is  the  distortion  of  the  orbits  as  pro- 
jected, we  may  compare  the  length  of  the  radius  vector  to  any 
point  in  the  orbit  with  that  of  its  projection  on  the  ecliptic,  which 
is  called  the  "  curtate  "  distance  from  the  sun. 

Even  in  the  case  of  the  orbit  of  Mercury,  which  has  the  greatest 
inclination,  the  curtate  distance  differs  from  the  true  distance  at 
most  by  y^,  in  the  case  of  Venus  by  less  than  ff  J^,  and  in  the 
case  of  all  the  other  planets  by  less  than  y^1^-  If  the  scale  of  the 
diagram  is  such  that  one  astronomical  unit  equals  1|  inches,  no 
radius  vector  drawn  in  any  one  of  the  "  curtate  "  orbits  will  differ 
from  the  corresponding  radius  vector  drawn  in  the  actual  orbit  by 
so  much  as  ^1^  of  an  inch  ;  and  by  referring  to  the  data  given  on 
page  146  it  will  be  seen  that  on  that  scale  the  elliptic  orbits  may 
be  represented  with  considerable  accuracy  as  circles. 

The  position  of  the  line  of  apsides  is  fixed  by  the  longitude  of 
perihelion,  page  174;  the  distance  c  of  the  center  of  the  ellipse 
from  the  sun  is  found  from  the  ratio  c  /a  =  e,  and  a  circle  struck 
about  the  center  with  a  radius  a  very  closely  represents  the  curtate 
orbit ;  the  distances  c  and  a  are  of  course  to  be  laid  off  from  the 
scale  of  astronomical  units. 

To  draw  such  a  diagram  is  a  useful  exercise,  and  by  careful  draw- 
ing and  erasure  a  single  diagram  may  serve  for  many  problems,  but 
it  is  convenient  to  have  several  printed  copies  when  it  is  desired  to 
preserve  the  solutions. 

It  is  also  convenient  to  have  diagrams  on  which  an  astronomical 
unit  equals  2j,  j,  and  f  inches,  respectively,  the  first  extending  to  the 
orbit  of  Mars,  the  second  to  that  of  Jupiter,  and  the  third  to  that 
of  Saturn.  The  larger  scale  should  be  used  for  problems  referring 
to  Mercury  and  Venus,  while  the  smaller  scales  are  required  for 
the  major  planets. 


152 


LABORATORY  ASTRONOMY 


ELEMENTS  OF  THE  SIX  INNER  PLANETS,  JAN.  I,  1900 


Sj~M 

S 

| 

e 

t 

y 

h 

MUD  DUUnco 

0.887 

0.72S 

1.000 

1.524 

6.203 

9.639 

Eoetttridt, 

0.2056 

0.0068 

0.0168 

0.0933 

0.0482 

0.0561 

Inclination 

7*0 

3*.4 

1*9 

1*3 

2*5 

Longitad.    of    A.- 
eeoduigNod. 

17*1 

76*.7 

48*7 

99*4 

U2*7 

Longitod.  of  Peri- 

helion 

76*« 

1304 

101*2 

S34!2 

12*7 

9o!tt 

Meu  Longitude, 
Gr.  Meu  Nooa 

182?22 

344*33 

100*67 

294*27 

238*13 

266*61 

Sidereal  Period, 
lieu  Solar  D»y> 

87*9693 

224-701 

865*266 

686*979 

4332*58 

10759*2 

Mean  dailj  motion 

4*09234 

1*60213 

0*98661 

o!62403 

0*08309 

0?03346 

EQUATION  OF  CENTER 


Mm, 

A°°"*" 

A*.* 

0* 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

860 

10 

5.4 

0.1 

tj 

2.1 

1.0 

1.2 

350 

10.5 

0.3 

0.7 

4.1 

2.0 

2.4 

340 

14.9 

0.4 

1.0 

6.9 

2.9 

8.4 

830 

18.6 

0.5 

1.2 

7.5 

3.7 

4.4 

820 

21.1 

0.0 

.5 

8.8 

4.4 

6.2 

310 

22.8 

0.7 

.7 

9.8 

4.9 

5.8 

300 

23.6 

0.7 

.8 

10.4 

5.3 

6. 

290 

23.6 

0.8 

.9 

10.7 

5.5 

6. 

280 

22.9 

M 

.9 

10.6 

u 

6. 

270 

21.7 

0.8 

19 

10.2 

M 

6. 

260 

19.9 

0.7 

.8 

0.6 

5.1 

5. 

250 

15.3 

o.n 

1.4 

7.6 

4 

4.7 

230 

12.5 

0.5 

1.2 

6.8 

S. 

3.9 

220 

50 

9.6 

0.4 

0.9 

4.8 

2 

3.0 

210 

60 

6.5 

0.3 

0.8 

S.S 

1. 

2.1 

200 

70 

3.3 

0.1 

0.3 

1.7 

0. 

1.0 

190 

80 

0.0 

0.0 

0.0 

0.0 

0. 

0.0 

180 

FIG.  80.    Diagram  of  Curtate  Orbits 


THE  MOTIONS  OF  THE  PLANETS  153 

On  the  plan  of  each  orbit  the  symbol  of  the  planet  is  placed  at 
the  perihelion  point,  whose  position  is  thus  approximately  known 
at  a  glance. 

That  part  of  the  orbit  which  is  above  the  plane  of  the  ecliptic  is 
marked  with  a  full  line,  and  the  part  below  is  marked  by  a  broken 
line.  The  line  of  nodes  is  therefore  determined  as  a  line  joining 
the  two  points  where  the  character  of  the  line  changes.  This  line, 
of  course,  passes  through  the  sun. 

The  inclinations  of  the  orbit  planes  are  shown  by  the  triangles 
which  appear  below  the  diagram,  each  marked  by  the  symbol  of 
the  planet  to  whose  orbit  it  pertains.  A  scale  of  astronomical  units 
is  printed  at  the  bottom. 

The  attached  tables  (see  page  174)  give  the  values  of  the  elements 
of  each  orbit  and  certain  other  quantities  which  are  required  in 
finding  the  place  of  the  planet  in  its  orbit  at  a  given  time. 

Measurements  may  be  made  on  the  diagram  between  any  two 
points  by  laying  a  strip  of  paper  with  its  straight  edge  through 
the  points,  and  marking  the  edge  of  the  strip  opposite  each  point. 
By  laying  the  straight  edge  along  the  scale  the  distance  in  astro- 
nomical units  is  found.  Instead  of  the  paper  strip  a  pair  of  com- 
passes may  be  used. 

The  map  shows  the  orbits  as  they  would  be  seen  from  the  north 
side  of  the  ecliptic,  and  the  motions  of  the  planets  as  thus  seen  are 
always  counter-clockwise  about  the  sun.  The  plane  of  the  map  is 
that  of  the  ecliptic,  and  it  is  so  oriented  on  the  paper  that  hori- 
zontal lines  drawn  from  left  to  right  would  strike  the  celestial 
sphere  at  the  vernal  equinox.  Therefore  the  direction  which  on 
an  ordinary  terrestrial  map  would  be  east  on  this  map  is  toward 
longitude  zero ;  up  is  toward  longitude  90°,  down  toward  longitude 
270°,  and  the  direction  of  any  other  line  on  the  map  is  fixed  by 
determining  the  angle  which  it  makes  with  the  line  drawn  to  the 
vernal  equinox.  Thus,  the  line  in  Fig.  81  from  E  to  M  makes  an 
angle  of  45°  with  the  line  SR,  and  is  therefore  directed  ^toward 
longitude  45°,  and  EJ  is  directed  toward  longitude  260°.  By  draw- 
ing lines  through  the  sun  parallel  to  EM  and  EJ,  respectively,  the 
longitude  may  be  read  off  directly  on  the  circle  which  bounds  the 
diagram. 


154 


LABORATORY  ASTRONOMY 


1  _L 


0  1  Z          3  4          5          6 

FIG.  81.    Direction  of  a  Line  fixed  by  Longitude 

To  find  the  Elements  of  an  Orbit.  —  The  elements  of  the  planetary 
orbits  may  be  obtained  from  measurements  on  the  diagram.  These 
elements  are  as  follows : 

a    Semi-axis  major  of  the  ellipse  or  mean  distance. 

e    Eccentricity  of   the  ellipse  =  c/a,  where  c  is  distance  of 

focus  from  center. 

TT    Heliocentric  longitude  of  perihelion. 
££    Heliocentric  longitude  of  node. 
i     Inclination  of  plane  of  orbit  to  plane  of  the  ecliptic. 


THE  MOTIONS  OF  THE  PLANETS  155 

To  find  a  draw  a  straight  line  from  the  perihelion  point  of  the 
orbit  through  the  sun  to  cut  the  orbit  at  the  aphelion  point.  This 
is  the  line  of  apsides.  Measure  the  distance  from  perihelion 
to  aphelion  along  the  line  of  apsides  in  astronomical  units.  This 
gives  the  major  axis  of  the  ellipse,  one-half  of  which  is  the  value 
of  a. 

To  find  c,  bisect  the  major  axis  and  thus  fix  the  center  of  the 
ellipse.  The  distance  from  focus  to  center  may  then  be  measured 
in  astronomical  units.  This  is  the  value  of  c;  it  is  not  regarded 
as  one  of  the  elements,  since  it  is  fixed  by  the  values  a  and  e. 

To  find  e,  determine  c/a  from  the  above  measurements. 

To  find  TT,  prolong  the  line  of  apsides  through  the  perihelion 
point  ;  the  reading  at  the  point  where  it  cuts  the  graduated  circle 
is  the  longitude  of  perihelion. 

To  find  Q,  ,  prolong  the  line  of  nodes  through  the  point  where 
the  planet  moving  counter-clockwise  passes  from  the  dotted  por- 
tion of  the  orbit  to  the  full  line.  The  reading  at  the  point  where 
this  line  cuts  the  graduated  circle  is  the  longitude  of  the  ascending 
node. 

To  find  the  inclination  i,  measure  the  angle  of  the  proper  triangle 
by  a  protractor  ;  or,  more  accurately,  measure  the  altitude  h  and 
the  base  b  of  the  triangle  ;  h/b  is  equal  to  the  inclination  in  radians. 
57°.3  h/b  =  i  in  degrees. 

The  following  measurements  were  made  on  the  orbit  of  Jupiter  : 

Sun  to  perihelion  ...     .............       4.96 

Sun  to  aphelion     .....     ....     .......       5.42 

Major  axis    ......  ........     .....     10.38 

a     Semi-axis  a    ................      5.19 

Center  to  perihelion      .............       5.19 

Focus  to  perihelion       .............       4.96 

c      Center  to  focus  c      ....     ......     ....       0.23 


. 

a       5.19 

TT      The  line  of  apsides  cuts  the  circle  at  12°.  7. 
Q     The  line  to  ascending  node  cuts  the  circle  at  99°.4. 
i      The   altitude   of   the    triangle    is   0.13    and    the    base   5.43;    hence 
i  =  h/b  =  0.13/5.43  =  0.024  radian  =  57°.3  x  0.024  =  1°.37. 


156 


LABORATORY  ASTRONOMY 


PLACE  OF  THE  PLANET  IN  ITS  ORBIT 

If  the  heliocentric  longitude  of  a  planet  is  known,  it  may  be 
plotted  at  its  proper  place  on  the  diagram  by  drawing  a  line  from 
the  sun  to  that  division  of  the  graduated  circle  which  indicates  the 
given  longitude ;  the  intersection  of  this  line  with  the  orbit  gives 
the  required  place.  When,  for  instance,  the  heliocentric  longitude 
of  Jupiter  is  280,  the  intersection  falls  very  close  to  the  descend- 
ing node.  In  this  particular  case  the  place  of  the  planet  is  com- 
pletely known,  since  it  is  in  the  ecliptic.  Usually  the  planet  is 


FIG.  82.    The  Z  Coordinate 

many  millions  of  miles  from  the  ecliptic,  but  its  exact  distance  may 
be  easily  found  by  the  use  of  its  inclination  triangle. 

This  will  appear  by  consideration  of  Fig.  82,  which  represents  a 
diagram  in  which  the  orbit  of  Jupiter  has  been  cut  through  along 
the  heavy  line,  and  the  part  of  the  orbit  which  is  above  the  ecliptic 
turned  up  around  the  line  of  nodes  so  as  to  be  at  the  proper  incli- 
nation. The  exact  angle  is  insured  by  supporting  it  by  wedges 
having  the  proper  angle. 


THE  MOTIONS  OF  THE  PLANETS 


157 


The  height  of  the  planet  at  P  above  the  plane  of  the  ecliptic, 
which  we  shall  call  its  "  Z  coordinate,"  or  simply  Z,  is  evidently 
the  altitude  of  a  right-angled  triangle  whose  small  angle  is  i  (the 
inclination  of  the  orbit),  and  whose  base  is  the  line  drawn  from 
the  place  of  the  planet  on  the  diagram  to  the  line  of  nodes.  This 
line  (which  practically  equals  the  hypotenuse)  we  will  call  U. 

To  find  Z,  then,  it  is  sufficient  to  measure  U  on  the  diagram  and 
to  lay  off  the  same  distance  along  the  horizontal  side  of  the  incli- 
nation triangle.  The  vertical  line  drawn  to  the  hypotenuse  from 
the  point  thus  fixed  gives  the  length  of  Z  in  astronomical  units. 
A  far  more  accurate  method  is  to  make  use  of  the  obvious  relation 
Z/U  =  i  in  radians,  or  57.3  Z/U  =  i  in  degrees.  Thus,  for  Jupiter 
Z  =  U  x  1.3/57.3  =  0.023  U. 


TO   FIND   THE    TRUE   HELIOCENTRIC    LONGITUDE   OF   A 

PLANET 

To  find  the  true  position  of  any  planet  at  a  given  time  we  must 
first  know  its  mean  anomaly  at  that  time,  and  then,  by  applying 
the  equation  of  center, 
find  the  correspond- 
ing value  of  the  true 
anomaly  which  enables 
us  to  place  the  planet  Eo 
at  the  proper  position 
in  its  orbit. 

Thus,  if  the  earth's 
mean  anomaly  is  70°, 
we  find  by  the  table, 
page  174,  that  the  equa- 
tion of  center  is  + 1°.8, 
and  hence  the  true 
anomaly  is  71°.8.  Since  the  longitude  of  perihelion  is  101°.2,  the 
true  heliocentric  longitude  is  101°.2  +  71°.8,  or  173°.0,  and  this 
value  enables  us  to  plot  the  earth  in  its  proper  place  on  the 
diagram,  Fig.  83. 


158  LABORATORY  ASTRONOMY 

We  may  find  the  mean  anomaly  if  we  know  the  number  of  days 
elapsed  since  perihelion,  and  the  mean  daily  motion  along  the  orbit. 
The  fact  that  the  planets  move  very  nearly  in  the  ecliptic,  so  that  the 
motion  in  the  real  and  curtate  orbit  is  very  nearly  the  same,  makes 
it  easier  to  proceed  in  a  somewhat  different  manner,  as  follows  : 

In  the  Table  of  Elements  appended  to  the  chart  is  given  the  "mean 
daily  motion  w  (in  heliocentric  longitude),  which  is  found  by  divid- 
ing 360°  by  the  period  in  days.  This  quantity  enables  us  by  a  simple 
multiplication  to  find  the  mean  motion,  or  increase  in  heliocentric 
longitude  of  the  mean  planet  in  any  given  number  of  days. 

Knowing  the  mean  (heliocentric)  longitude  at  any  given  epoch, 
the  mean  longitude  at  any  later  date  is  found  by  addition  of  the 
mean  motion  in  the  elapsed  time.  The  Table  of  Elements  supplies 
the  necessary  "  longitude  at  the  epoch "  for  Greenwich  mean  noon, 
January  1,  1900. 

We  may  summarize  the  process  of  finding  the  planet's  true  helio- 
centric longitude  as  follows  : 

Let  E  be  the  longitude  at  the  epoch, 

t  "  "  elapsed  time  in  days, 

/u,  "  "  mean  daily  motion, 

TT  "  "  longitude  of  perihelion, 

M  "  "  mean  anomaly, 

v  "  "  true  anomaly, 

/  "  "  true  longitude  (heliocentric). 

First  find  the  mean  anomaly  at  the  time  t,  as  follows  : 

pi  =  Mean  motion  in  elapsed  time, 
E  -f-  fA.t  =  Mean  longitude  at  given  date, 
E  +  fjit  —  TT  =  Mean  anomaly. 

With  this  value  of  the  mean  anomaly  find  the  equation  of  center 
by  the  table,  and  since 

True  anomaly  =  Mean  anomaly  -f  Equation  of  center, 
or  v  =  E  +  pi  —  TT  +  Equation  of  center, 

and      True  longitude  =  v  -f-  TT,          we  have  directly 
True  longitude  =  E  +  ^t  +  Equation  of  center. 

The  form  of  the  computation  is  shown  in  the  solution  of  the 
following  problem  : 


THE  MOTIONS  OF  THE  PLANETS  159 

Find  the  true  place  of  Mars  and  the  earth  May  8,  1905,  at 
Greenwich,  midnight. 

The  elapsed  time  may  be  found  as  follows  : 

Gr.  Mean  Noon.     Jan.  1,  1900,  to  Jan.  1,  1901  365  days 

1902  365 

1903  365 

1904  365 

1905  366 
Jan.  1,  1905,  to  Feb.  1,  1905          31 

Mar.  1,  1905          28 
Apr.  1,  1905          31 
May  1,  1905          30 
Noon.    May  1  to  Midn.,  May  8,  1905  7.5 

Elapsed  time  =  1953.5  days. 

For  Mars         td  =  0°.  52403  x  1953.5  =  1023°.  69. 
For  the  earth  &  =  0°.98561  x  1953.5  =  1925°.39. 

MARS  EARTH 

Mean  longitude  Jan.  1,  1900  =  E 294°. 27  100°. 67 

Mean  motion  1953.5  days  =  nt 1023  .69  1925  .39 

E  +  fit 1317  .96  2026  .06 

Subtract  complete  revolutions     ......  1080  .  1800  . 

Mean  longitude  May  8.5,  1905    ......  237  .96  226  .06 

Subtract  longitude  of  perihelion  ir [~  334.2  101.2  ~| 

Mean  anomaly  M |_  263.8  124.9  J 

Equation  of  center    .     .     .     .     .     .     .     .     .     .  -  10.36  +  1.50 

True  longitude  /  =  E  +  id  +  Equation  of  center  227.60  227.56 

It  will  be  noted  that  in  each  case  the  value  of  E  +  ^t  has  been 
diminished  by  an  integral  number  of  revolutions :  3  x  360°  for 
Mars  and  '5  x  360°  for  the  earth.  It  appears,  also,  that  the  num- 
bers inclosed  in  brackets  enter  the  computation  only  for  the  purpose 
of  obtaining  the  equation  of  center  which  is  then  applied  directly 
to  the  mean  longitude  following  the  equation 

I  =  E  +  fjit  +  equation  of  center. 

On  plotting  the  planets  it  appears  that  Mars  is  exactly  opposite 
the  sun,  as  indeed  is  evident  from  the  fact  that  the  earth  and  Mars 
are  in  the  same  heliocentric  longitude.  The  Ephemeris  gives  May  8, 
8  P.M.,  G.M.T.,  as  the  time  of  opposition.  The  actual  distance 
between  Mars  and  the  earth,  as  measured  on  the  diagram,  is  0.56 
astronomical  units,  or  fifty-two  million  miles. 


160 


LABORATORY  ASTRONOMY 


The  planet  may  be  plotted  with  a  very  fine-pointed,  hard  pencil, 
against  the  edge  of  a  ruler  passing  through  the  sun  and  the  point  of 
the  graduated  circle  whose  reading  equals  the  planet's  true  helio- 
centric longitude.  It  is  quite  an  advantage  to  have  the  ruler  of  a 
transparent  substance  in  order  that  its  edge  may  be  correctly  placed 
on  the  graduations. 

A  better  method,  however,  is  to  put  a  pin  through  the  sun's  place 
firmly  into  the  drawing  board  or  table,  and  pass  around  the  pin  a 
long  loop  of  smooth  black  thread.  The  other  end  of  the  loop  is 


FIG.  84.    Plotting  with  a  Loop 

held  between  the  thumb  and  forefinger,  with  the  threads  slightly 
separated  (about  ^  of  an  inch).  The  loop  is  then  drawn  taut,  and 
the  middle  of  the  white  space  between  the  threads  may  be  bisected 
by  the  proper  point  on  the  graduation ;  the  place  of  the  planet  is 
then  marked  by  putting  the  point  of  the  pencil  exactly  midway 
between  the  threads  where  they  intersect  the  orbit  (Fig.  84). 

The  planet  having  been  placed  in  its  true  position  on  the  orbit 
by  plotting  it  as  above,  so  that  its  curtate  radius  vector  is  drawn 
toward  the  true  heliocentric  longitude,  its  place  is  completely  known 
if  we  measure  U  and  find  Z,  as  on  page  157.  The  usual  method  of 
fixing  the  distance  of  the  planet  from  the  ecliptic  is  to  give  its 
heliocentric  latitude  b,  or  angular  distance  from  the  ecliptic,  which 
may  be  found  thus  (Fig.  85)  : 


THE  MOTIONS  OF  THE  PLANETS 


161 


b  =  Z/r  =  angular  distance  (radians)  of  planet  above  ecliptic  as 
seen  from  the  sun.  Combining  this  with  Z  =  U  x  i  (radians),  as 
explained  on  page  157, 

b  (radians)  =  —  i  (radians)  ; 

and  turning  each  side  of  the  equation 
into  degrees  by  multiplying  by  57.3,  we 
have 

(57.36)°=-  x  (57.3  i)°, 


or 


The  inclinations  are  so  small  that  the 
latitude  is  always  well  determined  by 
this  method. 


FIG.  85.    Heliocentric  Latitude 


GEOCENTRIC  POSITIONS 

When  a  planet  has  been  placed  on  the  diagram  by  its  heliocentric 
coordinates,  we  may  find  its  position  as  seen  from  the  earth ;  that 
is,  we  may  find  the  longitude  and  latitude  of  that  point  of  the 
celestial  sphere  upon  which  it  is  seen  projected  by  an  observer 
upon  the  earth. 

The  line  drawn  from  the  earth  to  the  planet  is  called  the  "line  of 
sight,"  and  its  projection  on  the  ecliptic  is  the  line  from  the  earth 
to  the  planet  on  the  diagram.  If  this  line  is  horizontal,  it  cuts 
the  celestial  sphere  at  the  vernal  equinox,  and  the  planet's  geocen- 
tric longitude  is  zero. 

Geocentric  Longitude. — The  angle  between  the  (projected)  line  of 
sight  and  the  line  drawn  to  the  vernal  equinox  is  the  planet's  geo- 
centric longitude.  It  is  equal  to  the  angle  between  the  line  of  sight 
and  the  line  drawn  from  the  sun  to  the  zero  of  the  graduated  circle. 
This  angle  may  be  measured  in  several  ways : 

1.  By  prolonging  the  line  of  sight,  if  necessary,  till  it  cuts  the 
line  of  equinoxes  on  the  diagram,  and  measuring  the  angle  with  a 
protractor. 


162 


LABORATORY  ASTRONOMY 


2.  By  drawing  a  line  through  the  sun   parallel  to  the  line  of 
sight,  and  noting  the  point  where  it  cuts  the  graduated  circle. 

3.  The  most  accurate  method  is  usually  the  following:  Bring  a 
straight  edge  to  pass  accurately  through  the  places  of  earth  and 
planet.    Note  the  points  of  intersection  with  the  graduated  circle. 


FIG.  86.    Geocentric  Longitude 


Call  the  reading  where  the  line  of  sight  (from  earth  to  planet) 
cuts  the  circle  A,  and  the  other  (opposite)  reading  B.    Then  the 


geocentric  longitude  of  the  planet  is 


A+B 


—  90,  if  A  is  less  than 


B ;  and \-  90,  if  A  is  greater  than  B.    This  may  be  proved  by 

2 

the  theorem  that  the  angle  between  two  chords  of  a  circle  is  meas- 
ured by  the  half  sum  or  half  difference  of  the  included  angles, 
according  as  they  intersect  inside  or  outside  the  circle. 

Better  than  a  straight  edge  is  a  fine  line  on  a  transparent  ruler 
(celluloid,  glass,  mica,  tracing  cloth),  or  a  stretched  thread  laid  over 
the  two  points. 

Fig.  86  illustrates  the  three  methods,  the  heliocentric  longitudes 
of  the  earth  and  Venus  being  150°  and  90°,  respectively.  The  angle 
at  C  measured  by  the  protractor  is  13°,  the  line  through  S  parallel 
to  AB  cuts  the  graduated  circle  at  13.0,  while  the  readings  at  A 

and  B  are  20.0  and  186.0,  so  that  —  —  -  90°  =  13°.0. 

The  Sun's  Longitude  and  the  Equation  of  Time.  — It  is  an  important 
fact  that,  since  the  line  of  sight  to  the  sun  is  drawn  to  a  point 


THE  MOTIONS  OF  THE  PLANETS  163 

whose  heliocentric  longitude  is  opposite  to  that  of  the  earth,  the 
sun's  geocentric  longitude  is  always  180°  +  the  earth's  heliocentric 
longitude. 

The  sun  appears  to  move  about  the  earth  in  an  orbit  whose  ele- 
ments are  the  same  as  those  of  the  earth  about  the  sun,  except 
that  E  and  TT  are  each  greater  by  180°. 

The  sun's  mean  longitude  is  therefore  280°.67  -f-  fit  and  its  mean 
anomaly  is  280°.67  -f  /*£  —  281°.2,  where  t  is.  the  number  of  days 
since  January  1,  1900,  and  /A  is  the  earth's  mean  daily  motion. 

To  find  the  sun's  true  longitude  we  add  to  the  mean  longitude 
the  equation  of  center  taken  from  the  table  for  the  earth,  and  from 
the  true  longitude  we  may  find  the  R.A.  by  adding  the  reduction 
to  the  equator  (page  121).  We  may  therefore  write : 

Sun's  R.A.  =  Sun's  mean  longitude  +  Eq.  center  -f-  Red.  to  equator. 
Sun's  R.A.  —  Sun's  mean  longitude  =  Eq.  center  +  Red.  to  equator. 

And  since  the  sun's  mean  longitude  equals  the  R.A.  of  the  mean 
sun  (page  92), 

Sun's  R.A.  —  R.A.  of  mean  sun  =  Eq.  center  -f-  Red.  to  equator. 

The  first  member  of  the  last  equation  is  the  equation  of  time 
, whose  approximate  value  may  thus  be  computed  for  any  date : 

Jan.  31,  1900.       ^t  =  30  x  0°.9S56  =    29°.57 

E  +  pt  =  280°.67  +  29°.57  =  310  .24 

E  +  iLt-Tr  =  310°.24  -  281°.2  =    29  .04 

Equation  of  center  =  +  0  .97          +  0°.97 
True  longitude  311°.21 

Red.  to  equator  +  2  .4 

Equation  of  time  +  3°.37 

or  13.5  minutes  to  be  added  to  apparent  time. 

Geocentric  Latitude. — The  geocentric  latitude  ft  of  the  planet  is 
the  angular  distance  of  the  planet  from  the  ecliptic  as  seen  from  the 
earth.  It  is  found  by  the  same  method  as  that  used  for  finding  the 
heliocentric  latitude  b.  (See  Fig.  87.) 

Draw  the  line  A  from  earth  to  planet  on  the  diagram.  Z/A 
equals  the  angle  (3  in  radians,  and  Z  —  U  x  i  (in  radians). 


164  LABORATORY  ASTRONOMY 

Hence,  by  reasoning  applied  on  page  161, 


FlG.  87.    Geocentric  Latitude 

The  whole  process  of  finding  geocentric  latitude  and  longitude 
is  illustrated  in  the  following  example  : 

To  find  the  positions  of  the  five  inner  planets  at  Greenwich 
mean  noon,  July  6,  1907,  the  elapsed  time  from  January  1, 
1900,  being  2742  days  (see  page  167). 


$ 

9 

0 

d1 

3 

E     

182°.22 

344°.33 

100°.67 

294°  27 

238.13 

Add  fit       ..'.... 

11221  .20 

4393  .04 

2702  .54 

1436  .89 

227.83 

E  4-  id 

11403   42 

4737    37 

2803   21 

1731    16 

465  96 

Subtr.  complete  revolutions 

11160  . 

4680  . 

2520  . 

1440  . 

360. 

E  -f  fit       

243  .42 

57  .37 

283  .21 

291  .16 

105.96 

Subtract  TT     

75  .9 

130  .2 

101  .2 

334  .2 

12.7 

Mean  anomaly    .... 
Eq.  of  center  by  table 
Heliocentric  longitude  I    . 

167  .5 
+  4  .10 
247  .52 

287  .2 
-0  .73 
56  .64 

182  .0 
-0  .06 
283  .15 

317  .0 

-7  .89 
283  .27 

93.3 
+  5.47 
111.43 

Plotting  the  planets  on  the  diagram,  we  determine  the  geocentric 
places  by  finding  the  following  values  : 


THE  MOTIONS  OF  THE  PLANETS 


165 


« 

9 

* 

i 

From 

diagram, 

A           .     .          .     . 

130°.  4 

80°.  1 

283°.  2 

111°  2 

u 

u 

B 

302°  3 

266°  9 

103°.  3 

288°  8 

„ 

u 

U      

0.14 

0.22 

1.16 

1.09 

u 

From 

u 

table 

A       

i 

0.70 

7°0 

1.60 

3°.  4 

0.40 
1°  9 

6.26 
1°  3 

A  +  J 

3 

±90 

126°  35 

83°  5 

283°  25 

110°  0 

2 
U 

—  X  I 

—  8     . 

-  0°.47 

-  5°.  51 

+  0°.21 

A 

FIG.  88.    Geocentric  Places,  July  6,  1907 


166 


LABORATORY  ASTRONOMY 


The  signs  attached  to  the  latitudes  are  fixed  by  the  fact  that 
Jupiter  is  in  the  full-line  part  of  its  orbit  and  therefore  above  the 
ecliptic,  while  all  the  other  planets  are  in  the  dotted  parts  of  their 
orbits  and  therefore  in  south  latitudes. 

Since  the  full  line  extends  from  the  longitude  of  the  ascending 
node  to  that  of  the  descending  node,  which  is  180°  greater,  we  may 
also  fix  the  sign  of  /3  by  the  following  rule  : 

From  the  true  heliocentric  longitude  subtract  that  of  the  ascend- 
ing node ;  if  I  —  Q>  <  180,  the  latitude  is  positive  ;  if  I  —  &  >  180, 
the  latitude  is  negative.  Thus,  in  the  above  example  : 


$ 

9 

cf 

y 

1    

247°.  5 

56°.  6 

283°.  3 

111°.4 

Q    

47  .1 

75.7 

48.7 

100.1 

I  _  o     

200  .4 

340  .9 

234  .6 

11  .3 

B   . 

nesr. 

near. 

nesr. 

pos. 

Perturbations.  —  The  longitudes  above  obtained  are  liable  to  an 
error  of  more  than  a  tenth  of  a  degree  if  the  elapsed  time  exceeds 
a  half  century,  and  the  perturbations  which  are  neglected  may  add 
somewhat  to  the  error.  The  effect  of  the  mutual  perturbations  of 
Jupiter  and  Saturn  may  be  approximately  corrected  by  adding  to 
the  mean  longitudes  the  following  quantities  : 


3 

h 

h 

1800-1890 

+  0°.3 

1800-1840 

-0°.8 

1940-1960 

-0°.4 

1890-1950 

+  0  .2 

1840-1870 

-0  .7 

1960-1980 

-0  .3 

1950-1990 

+  0  .1 

1870-1910 

-0  .6 

1980-1990 

-0  .2 

1990-2000 

±0  .0 

1910-1940 

-0.5 

1990-2000 

-0  .1 

Effect  of  Precession.  —  The  true  longitudes  found  by  the  method 
above  described  are  referred  to  the  equinox  of  1900,  the  point  from 
which  the  mean  longitude  of  the  table  is  measured. 

Since  the  vernal  equinox  moves  along  the  ecliptic  50"  per  year 
toward  the  west,  or  nearly  6'  in  seven  years,  the  longitudes  meas- 
ured from  the  true  equinox  of  1907  will  be  about  0°.l  greater  than 


THE  MOTIONS  OF  THE  PLANETS  167 

if  measured  from  the  equinox  of  1900.  This  "reduction  to  the 
equinox  of  date"  is  50"  x  t,  or  0°.014  t,  where  t  is  the  number  of 
years  elapsed  since  1900. 

The  Julian  Day.  —  The  process  of  computing  the  elapsed  time  used 
on  page  159  is  tedious  and  liable  to  error  where  the  elapsed  time 
is  considerable.  Where  the  interval  between  distant  dates  is  to  be 
accurately  determined  astronomers  find  it  convenient  to  make  use 
of  the  number  of  each  day  in  the  Julian  period.  It  is  sufficient 
here  to  say  that  January  1,  4713  B.C.,  was  the  first  day  of  this 
period,  and  the  Ephemeris  gives  each  year  the  number  of  the 
Julian  day  for  January  1 ;  thus,  the  1st  of  January,  1900,  was 
No.  2415021  in  the  cycle.  To  find  the  number  for  any  given 
date,  we  turn  to  page  III  of  the  corresponding  month,  add  the 
day  of  the  year  (taken  from  the  second  column),  and  subtract  1. 

The  table  on  page  175  gives  for  each  year  from  1800  to  2000  a 
number  one  less  than  that  of  the  Julian  day  corresponding  to  Jan- 
uary 1  of  the  given  year.  The  subsidiary  table  for  months  gives 
for  each  month  a  number  one  less  than  the  day  of  the  year  cor- 
responding to  the  first  of  the  given  month. 

It  is  easy  to  see  that  by  adding  together  the  year  number,  month 
number,  and  day  of  the  month,  we  get  the  corresponding  Julian 
day.  Thus  we  compute  the  interval  from  January  1, 1900,  to  July  6, 
1907,  as  follows  : 

Year  number  for  1900,  2415020  1907  2417576 

Month  number  for  January,  0  _  July  181 

Day  of  month,  1  6  6 

Julian  day,  2415021  2417763 

2415021 
Elapsed  time,  2742 

Right  Ascensions  and  Declinations  of  the  Planets. — By  means  of  the 
geocentric  latitudes  and  longitudes  which  we  have  thus  determined 
the  planets  may  be  placed  in  their  respective  positions  upon  the  globe. 

The  proper  longitude  being  found  upon  the  ecliptic  of  the  globe, 
the  latitude  is  laid  off  on  a  strip  of  paper  by  placing  it  along  the 
ecliptic  and  marking  off  the  proper  number  of  degrees  along  its 
edge.  The  paper  is  then  applied  to  the  globe  so  as  to  mark  off  this 
distance  perpendicular  to  the  ecliptic.  The  latitude  is  never  so 


168  LABORATORY  ASTRONOMY 

great  as  8°,  so  that  no  serious  error  in  the  place  will  occur  if  the 
strip  is  not  exactly  perpendicular  to  the  ecliptic. 

The  place  of  the  planet  being  thus  marked  on  the  globe,  its  right 
ascension  and  declination  may  be  determined,  and  problems  relat- 
ing to  its  diurnal  motion,  such  as  its  times  of  rising  and  setting, 
may  be  solved  by  the  methods  of  Chapters  VIII  and  IX. 

CONFIGURATIONS  OF  THE   PLANETS 

The  elongation  of  a  planet  is  its  distance  from  the  sun  along  the 
ecliptic  as  seen  from  the  earth.  It  is  therefore  equal  to  the  differ- 
ence of  the  geocentric  longitudes  of  the  sun  and  planet.  The  elonga- 
tion is  measured  either  way  from  the  sun  up  to  180°,  at  which 
point  the  planet  is  at  opposition,  or  opposite  the  sun.  When  the 
elongation  is  zero  the  sun  and  planet  are  in  the  same  longitude, 
and  the  planet  is  in  conjunction  with  the  sun. 

The  symbols  §  and  <$  are  used  for  opposition  and  conjunction, 
respectively.  When  the  longitude  of  the  planet  is  greater  than 
that  of  the  sun  it  is  east  of  the  latter,  and  follows  it  in  its  diurnal 
revolution.  It  is  therefore  above  the  horizon  at  sunset  and  is 
said  to  be  an  "evening  star,'7  since  it  is  visible  in  the  twilight 
after  sunset  except  when  near  conjunction.  On  the  other  hand,  all 
planets  whose  longitudes  are  less  than  that  of  the  sun  precede  it, 
and  they  will  be  above  the  horizon  at  sunrise  and  therefore  visible 
at  dawn,  except  when  very  near  conjunction.  They  are  then  "  morn- 
ing stars,"  just  as  stars  in  eastern  elongation  are  evening  stars. 

The  geocentric  longitude  of  the  sun,  July  6,  1907,  is  103°.2  (since 
the  earth's  heliocentric  longitude  is  283°. 2,  page  164).  The  longi- 
tude of  Jupiter  being  110°. 2,  its  elongation  is  about  7°  east,  and  it 
is  an  evening  star,  though  too  close  to  the  sun  to  be  visible ;  it  will 
become  a  morning  star  about  July  14. 

The  elongation  of  Mars  is  very  nearly  180°,  and  it  is  at  opposi- 
tion and  becoming  an  evening  star.  The  longitude  of  Venus  is  84°.6 ; 
it  is  18°.6  west  of  the  sun  and  is  a  morning  star.  On  referring  to 
the  diagram  (Fig.  88),  and  remembering  that  it  moves  more  rapidly 
than  the  earth,  it  is  evident  that  it  is  approaching  conjunction 
beyond  the  sun  (" superior"  conjunction),  after  which  it  will  pass 


THE  MOTIONS  OF  THE  PLANETS  169 

to  eastern  elongations  and  be  an  evening  star.  Mercury's  longitude 
is  126° ;  it  is  23°  east  of  the  sun,  and  referring  to  the  diagram,  we 
see  that  it  is  approaching  conjunction  between  the  earth  and  sun 
("  inferior  "  conjunction),  after  which  it  will  be  a  morning  star. 


The  preceding  principles  enable  us  to  find  the  place  of  a  planet 
at  any  given  date,  and  thus  to  answer  many  of  the  questions  which 
continually  suggest  themselves  to  one  interested  in  watching  the 
courses  of  the  planets  in  the  sky. 

It  is  evident,  for  instance,  from  the  problems  solved  on  pages  159 
and  164,  that  in  1907  the  greater  proximity  of  Mars  to  the  earth 
offers  conditions  for  the  study  of  its  surface  which  are  much  more 
favorable  than  those  of  the  opposition  of  1905. 

The  oppositions  of  Mars  recur  at  an  average  interval  of  about 
780  days,  which  is  the  synodic  period  of  the  earth  and  Mars,  as 
explained  in  the  text-books  of  descriptive  astronomy. 

We  may  fix  the  dates  of  other  oppositions  approximately,  as  in 
August,  1877,  September,  1909,  November,  1911,  December,  1913, 
etc.,  and  by  computing  for  the  first  and  last  days  of  those  months  a 
closer  approximation  to  the  day  of  opposition  may  quickly  be  made, 
and  finally  a  careful  computation  for  the  exact  date  will  fix  the 
time  within  a  few  hours.  The  geocentric  place  and  the  distance  of 
the  planet  may  then  be  found. 

It  appears  that  favorable  oppositions  occur  in  the  summer,  and 
that  the  planet  is  then  quite  a  distance  south  of  the  equator,  so 
that  it  is  far  from  the  zenith  of  any  northern  observatory. 

The  satellites  of  Mars  were  discovered  in  1877,  and  in  the  same 
year  an  expedition  was  sent  to  the  island  of  Ascension  to  observe 
Mars  for  a  determination  of  the  solar  parallax. 

In  conclusion  we  will  consider  the  motion  of  Mars  during  the 
summer  of  1907,  to  illustrate  the  form  which,  the  computation  takes 
when  many  places  are  to  be  found  at  comparatively  short  intervals. 

We  first  carefully  determine  the  mean  longitudes  of  Mars  and 
the  earth  for  March  22  to  be  235°.51  and  178°.74,  respectively,  and 
then  easily  form  the  second  column  of  the  following  schedule  by 
successive  additions  of  10°.48  and  19°.71,  the  mean  motions  of  the 
two  planets  in  twenty  days. 


170 


LABORATORY  ASTRONOMY 


The  third  column  is  formed  for  Mars  by  writing  the  longitude 
of  perihelion  334°. 2  on  the  upper  edge  of  a  slip  of  paper  and 
placing  it  under  the  numbers  of  the  second  column  successively, 
subtracting  from  each  to  find  the  corresponding  mean  anomaly. 

The  same  result  is  more  easily  obtained  by  adding  in  the  same 
way  25°.8  (360°-334°.2)  to  each  number  in  the  second  column. 
The  third  column  is  checked  by  noting  that  the  differences  of  the 
successive  values  are  10°.48,  which  insures  the  accuracy  of  both 
columns.  The  equation  of  center  is  taken  from  the  table  and 
entered  in  the  fourth  column,  and  the  true  heliocentric  longitude 
found  by  adding  corresponding  numbers  of  the  second  and  fourth 
columns.  The  same  process  gives  the  earth's  true  heliocentric 
longitude. 

The  labor  is  by  no  means  proportionate  to  that  required  in 
computing  a  single  place,  and  the  comparison  of  the  successive 
numbers  of  each  column  is  an  important  aid  in  detecting  errors. 


MARS 

THE  EARTH 

Date 

E  +  fj.t 

E  +  u.t-n 

Eq.  of 

Center 

I 

E  +  nt 

E+H'-TT 

Eq.  of 
Center 

I 

Mar.  22 

235°.  51 

261°.3 

-  10°.3 

225°.  2 

178°.  74 

77°.5 

+  1°.9 

180°.  6 

April  11 

245  .99 

271  .8 

-10  .6 

235  .4 

198  .45 

97  .2 

+  1  .9 

200  .3 

May  1 

256  .47 

282  .3 

-10  .6 

245  .9 

218  .16 

117  .0 

+  1  -7 

219  .9 

21 

266  .95 

292  .8 

-10  .3 

256  .6 

237  .87 

136  .6 

+  1.3 

239  .2 

June  10 

277  .43 

303  .2 

-9.5 

267  .9 

257  .58 

156  .4 

+  0.7 

258  .3 

30 

287  .91 

313  .7 

-8.3 

279  .6 

277  .29 

176  .1 

+  0.1 

277  .4 

July  20 

298  .39 

324  .2 

-  6  .8 

291  .6 

297  .00 

195  .8 

-  0  .5 

296  .5 

Aug.  9 

308  .87 

334  .7 

-  5  .0 

303  .9 

316  .71 

215  .5 

-1  .1 

315  .6 

29 

319  .35 

345  .1 

-  3  .1 

316  .3 

336  .42 

235  .2 

-1  .5 

334  .9 

Sept.  18 

329  .83 

355  .6 

-0.9 

328  .9 

356  .13 

254  .9 

-1.8 

354  .3 

Oct.  7 

340  .31 

6  .1 

+  1.3 

311  .6 

15  .84 

274  .6 

-1  .9 

13  .9 

The  planets  were  plotted  from  the  above  data  on  a  scale  of  1.6 
inches  to  the  astronomical  unit,  the  boundary  circle  being  9| 
inches  in  diameter.  The  values  of  A,  B,  U,  and  A  were  determined 
and  the  geocentric  longitudes  and  latitudes  A  and  ft  found  as  in 
the  following  table : 


THE  MOTIONS  OF  THE  PLANETS 


171 


A 

B 

U 

A 

A. 

/3 

March  22  .  . 

244°.  6 

103°.  7 

+  0.13 

1.10 

264°.  1 

+  0°.2 

April  11  .  . 

254  .8 

112  .6 

-0.16 

0.91 

273  .7 

-0  .3 

May   1  .  . 

263  .9 

119  .2 

0.44 

0.75 

281  .5 

-  1  .2 

21  .  . 

271  .4 

120  .9 

0.69 

0.60 

286  .1 

-2  .2 

June  10  . 

277  .8 

117  .1 

0.90 

0.50 

287.4 

-3  .4 

30  .  . 

282  .2 

108  .0 

1.10 

0.44 

285  .0 

-4  .8 

July  20  .  . 

285  .7 

94.3 

1.25 

0.42 

280  .0 

-5  .7 

Aug.   9  .  . 

289  .7 

84  .9 

1.36 

0.47 

277  .3 

-5  .5 

29  .  . 

296  .8 

84.1 

1.39 

0.54 

280.4 

-4  .9 

Sept.  18  .  . 

305  .0 

88.2 

1.36 

0.64 

286  .6 

-4  .0 

Oct.   7  .  . 

316  .6 

98  .4 

1.27 

0.76 

297  .5 

-3  .2 

310  300          290          280          270  260          250          240 


FIG.  89.    Path  of  Mars  in  the  Summer  of  1907 


172  LABORATORY  ASTRONOMY 

In  order  to  form  an  idea  of  the  path  described  by  the  planet 
among  the  stars,  the  positions  may  be  plotted  on  an  ecliptic  map,  as 
in  Fig.  89,  which  shows  the  form  of  the  loop  in  the  constellation 
of  Sagittarius. 

During  March  the  motion  of  the  planet  is  eastward,  or  in  the 
direction  of  increasing  longitudes,  and  is  said  to  be  "  direct."  The 
rate  of  motion  diminishes  from  one-half  degree  per  day  at  the  out- 
set to  half  that  amount  in  May,  and  soon  after  the  beginning  of 
June  the  planet  reaches  its  first  "  stationary  point "  and  begins  to 
move  slowly  in  the  opposite  direction  in  longitude,  or  "retrograde." 
Its  continuous  motion  in  latitude  toward  the  south  prevents  it  from 
exactly  retracing  its  path  and  causes  it  to  describe  a  "loop." 

Its  velocity  in  the  retrograde  arc  increases  to  a  maximum  of 
a  quarter  of  a  degree  per  day  at  opposition  early  in  July,  and 
then  decreases  until  the  second  stationary  point  is  reached  about 
August  9,  when  the  planet  resumes  its  direct  motion. 

The  exact  dates  of  the  stationary  points  may  be  found  by  com- 
puting a  few  places  in  the  neighborhood  of  June  10  and  August  9. 

The  Ephemeris  gives  the  dates  as  June  5  and  August  8. 


THE  MOTIONS  OF  THE  PLANETS 


173 


TABLE  III  —  AVERAGE  VALUES  OF  THE  SUN'S  LONGITUDE  AND 
THE  EQUATION  OF  TIME 


LONGITUDE 

LONGITUDE 

MEAN  LONGITUDE 

EQ.  OF  TIME 

Jan.  1  .  .  . 

280°.  3 

>J  10°.  3 

280°.  1 

+  3">.  5 

11  .  .  . 

290  .5 

>?  20  .5 

289  .9 

+  7  .9 

21  ... 

300  .7 

XX  30  .7 

299  .8 

+  11  .3 

31  ... 

310  .8 

SZS  10.8 

309  .6 

+  13  .6 

Feb.  10  ... 

320  .9 

SXH  20  .9 

319.5 

+  14  .4 

20  ... 

331  .1 

X  1  .1 

329  .4 

+  14  .0 

Mar.  2  ... 

341  .3 

X  11.3 

339  .5 

+  12  .4 

12  ... 

351  .4 

X  21  .4 

349  .3 

+  10  .0 

22  . 

1  .3 

°f>  1  .3 

359  .2 

+  7  .1 

April  1  ... 

11  .2 

°|°  11  .2 

9  .0 

+  4  .1 

11  . 

21  .0 

°P  21  .0 

18.9 

+  1  .2 

21  . 

30  .8 

8   0.8 

28  .7 

-  1  .2 

May  1  ... 

40  .6 

8  10.6 

38  .6 

-  2  .9 

11  ... 

50.3 

8  20.3 

48  .5 

-  3  .7 

21  ... 

59  .9 

8  29.9 

58  .3 

-  3  .6 

31  .  .  •  . 

69  .5 

n  9  .5 

68  .2 

-  2  .6 

June  10  ... 

79  .0 

n  19  .0 

78  .0 

-  0  .9 

20  ... 

88  .6 

n  28.6 

87  .9 

+  1  .2 

30  ... 

98  .1 

EB  8  .1 

97  .7 

+  3  .3 

July  10  .  .  . 

107  .7 

ZB  17  .7 

107  .6 

+  5  .0 

20  ... 

117  .2 

£5  27  .2 

117  .5 

+  6  .1 

30  ... 

126  .7 

£1  6.7 

127  .3 

+  6  .2 

Aug.  9  ... 

136  .3 

£1  16  .3 

137  .2 

+  5  .4 

19  ... 

145.9 

£1  25.9 

147  .0 

+  3  .6 

29  ... 

155  .6 

TTJ?   5  .6 

156  .9 

+  0  .9 

Sept.  8  ... 

165  .3 

H£  15  .3 

166  .7 

-  2  .3 

18  . 

175  .0 

-n^  25  .0 

176  .6 

-  5  .8 

28  ... 

184  .8 

±±  4  .8 

186  .5 

-  9  .2 

Oct.  8  ... 

194  .6 

:£:  14  .6 

196  .3 

-12  .3 

18  .  . 

204  .5 

£±  24.5 

206  .2 

-14  .7 

28  ... 

214  .5 

Til  4  .5 

216  .0 

-16  .1 

Nov.  7  ... 

224  .5 

TH.  14  .5 

225  .9 

-16  .2 

17  .  . 

234  .6 

m.  24  .6 

235  .7 

-  15  .0 

27  ... 

244  .7 

t   4.7 

245  .6 

-12  .4 

Dec.  7  .  .  . 

254  .8 

t  14  .8 

255  .4 

-  8  .5 

17  ... 

265  .0 

t  25  .0 

265  .3 

-  3  .9 

27  ... 

275  .2 

>?  5.2 

275  .2 

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Jan.  6  . 

285  .4 

V?  15.4 

285  .0 

+  5  .7 

174 


LABORATORY  ASTRONOMY 


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THE  MOTIONS  OF  THE  PLANETS 


175 


TABLE  VII  —  THE  JULIAN  DAY 

Add  together  the  year  number,  the  month  number,  and  the  day  of  the  month. 


1800  2378496 

1843  2394201 

1886  2409907 

1929  2425612 

1972  2441317 

1801   78861 

1844   94566 

1887   10272 

1930   25977 

1973   41683 

1802   79226 

1845   94932 

1888   10637 

1931   26342 

1974   42048 

1803   79591 

1846   95297 

1889   11003 

1932   26707 

1975   42413 

1804   79956 

1847   95662 

1890   11368 

1933   27073 

1976   42778 

1805   80322 

1848   96027 

1891   11733 

1934   27438 

1977   43144 

1806   80687 

1849   96393 

1892   12098 

1935   27803 

1978   43509 

1807   81052 

1850   96758 

1893   12464 

1936   28168 

1979   43874 

1808   81417 

1851   97123 

1894   12829 

1937   28534 

1980   44239 

1809   81783 

1852   97488 

1895   13194 

1938   28899 

1981   44605 

1810   82148 

1853   97854 

1896   13559 

1939   29264 

1982   44970 

1811   82513 

1854   98219 

1897   13925 

1940   29629 

1983   45335 

1812   82878 

1855   98584 

1898   14290 

1941   29995 

1984   45700 

1813   83244 

1856   98949 

1899   14655 

1942   30360 

1985   46066 

1814   83609 

1857   99315 

1900   15020 

1943   30725 

1986   46431 

1815   83974 

1858   99680 

1901   15385 

1944   31090 

1987   46796 

1816   84339 

1859  2400045 

1902   15750 

1945   31456 

1988   47161 

1817   84705 

1860   00410 

1903   16115 

1946   31821 

1989   47527 

1818   85070 

1861   00776 

1904   16480 

1947   32186 

1990   47892 

1819   85435 

1862   01141 

1905   16846 

1948   32551 

1991   48257 

1820   85800 

1863   01506 

1906   17211 

1949   32917 

1992   48622 

1821   86166 

1864   01871 

1907   17576 

1950   33282 

1993   48988 

1822   86531 

1865   02237 

1908   17941 

1951   33647 

1994   49353 

1823   86896 

1866   02602 

1909   18307 

1952   34012 

1995   49718 

1824   87261 

1867   02967 

1910   18672 

1953   34378 

1996   50083 

1825   87627 

1868   03332 

1911   19037 

1954   34743 

1997   50449 

1826   87992 

1869   03698 

1912   19402 

1955   35108 

1998   50814 

1827   88357 

1870   04063 

1913   19768 

1956   35473 

1999   51179 

1828   88722 

1871   04428 

1914   20133 

1957   35839 

2000   51544 

1829   89088 

1872   04793 

1915   20498 

1958   36204 

Month  Num. 

1830   89453 

1873   05159 

1916   20863 

1959   36569 

Common 

Lp. 

1831  '  89818 

1874   05524 

1917   21229 

1960   36934 

Jan.   0 

~0 

1832   90183 

1875   05889 

1918   21594 

1961   37300 

Feb.  31 

31 

1833   90549 

1876   06254 

1919   21959 

1962   37665 

March  59 

60 

1834   90914 

1877   06620 

1920   22324 

1963   38030 

April  90 

91 

1835   91279 

1878   06985 

1921   22690 

1964   38395 

May  120 

121 

1836   91644 

1879   07350 

1922   23055 

1965   38761 

June  151 

152 

1837   92010 

1880   07715 

1923   23420 

1966   39126 

July  181 

182 

1838   92375 

1881   08081 

1924   23785 

1967   39491 

Aug.  212 

213 

1839   92740 

1882   08446 

1925   24151 

1968   39856 

Sept.  243 

244 

1840   93105 

1883   08811 

1926   24516 

1969   40222 

Oct.  273 

274 

1841   93471 

1884   09176 

1927   24881 

1970   40587 

Nov.  304 

305 

1842   93836 

1885   09542 

1928   25246 

1971   40952 

Dec.  334 

335 

n. 


MARCH,  1899. 


AT  GREENWICH  MEAN  NOON. 

THE  SUN'S 

| 

| 

Equation  of 

Sidereal 

0) 

Time, 
to  be 

Time, 
or 

1 

•3 

Subtracted 

Right  Ascension 

•8 

*J 

Apparent 

Diff.  for 

Apparent 

Diff.  for 

from 

Diff.  for 

of 

S 

Q 

Right  Ascension. 

i  Hour. 

Declinatioa 

i  Hour. 

Mean  Time. 

i  Hour. 

Mean  .Sun. 

Wed. 

I 

h     m        s 
22    48    49.18 

s 

S.   7  33     9-5 

+56.99 

m       s 
12    31.64 

8 
0.496 

h      m        8 
22   36    17-54 

Thur. 

2 

22  52  33-57 

9-34° 

7  10  18.4 

57.26 

12    19.48 

O.5l6 

22   40    14.09 

Frid. 

3 

22    56    17.49 

9-320 

6  47  21.0 

57-51 

12      6.84 

0.536 

22  44  10.64 

Sat. 

4 

23    o    0.95 

9.301 

6  24  17.9 

+57-75 

ii  53-75 

0-555 

22   48      7.20 

SUN. 

5 

23      3    43-98 

9.284 

6     i     9.3 

57-97 

ii  40.23 

0.572 

22  52    3-75 

Mon. 

6 

23    7  26.60 

9.267 

5  37  55-6 

58.17 

ii  26.29 

0.589 

22    56      0.30 

Tues. 

7 

23  ii    8.82 

9-251 

5  H  37-3 

+58.35 

ii  11.96 

O.6O5 

22  59  56.86 

Wed. 

8 

23  14  50.66 

9.236 

4  5i  14-8 

58.52 

10  57.25 

O.62O 

23      3    53-41 

Thur. 

9 

23  18  32.14 

9.221 

4  27  48.4 

58.67 

10  42.17 

0.635 

23    7  49-96 

Frid. 

10 

23    22    13.27 

9.207 

4    4  18.6 

+58.80 

10  26.75 

0.649 

23  II  46.52 

Sat. 

ii 

23    25    54.08 

9.194 

3  4°  45-8 

58.92 

IO    II.OI 

0.663 

23  15  43-07 

SUN. 

12 

23  29  34-57 

9.181 

3  17  10.3 

59-02 

9  54-95 

0.675 

23  19  39-62 

Mon. 

13 

23  33  14.77 

9.169 

2  53  32-7 

+59-10 

9  38.60 

0.687 

23  23  36.17 

Tues. 

«4 

23  36  54-69 

9.158 

2    29    53.2 

59-17 

9  21.96 

0.698 

23  27  32.73 

Wed. 

23  4°  34-35 

9.147 

2     6  12.3. 

59-22 

9     5-°7 

0.709 

23  31  29.28 

Thur. 

16 

23  44  13.76 

9-137 

I    42   30-4 

+59.26 

8  47-93 

0.719 

23  35  25-83 

Frid. 

17 

23  47  52.95 

9.128 

I   18  47.8 

59.28 

8  30.56 

0.728 

23  39  22.38 

Sat. 

18 

23  51  31.93 

9.120 

o  55     4-9 

59.28 

8  12.99 

0.736 

23  43  18.94- 

SUN. 

19 

23  55  10-72 

9-"3 

0    31    22.2 

+59-27 

7  55-23 

0-744 

23  47  15-49 

Mon. 

20 

23  -58  49.34 

9.106 

S.   o    7  39.8 

59-25 

7  37-30 

0.750 

23  51  12.04 

Tues. 

21 

0      2    27.82 

9.100 

N.  o  16     1.7 

59-21 

7  19.22 

0.756 

23  55    8.60 

Wed. 

22 

o    6    6.17 

9.095 

o  39  42.1 

4.59.15 

7     1.02 

0.76l 

23  59    5-15 

Thur. 

23 

o    9  44.41 

9.092 

i     3  21.  o 

59-o8 

6  42.71 

0.765 

o    3     1.70 

Frid. 

24 

o  13  22.57 

9.089 

I  26  58.1 

59.00 

6  24.32 

0.767 

o    6  58.26 

Sat. 

25 

o  17    0.68 

9.087 

i  50  33-o 

+58.90 

6     5-87 

0.769 

o  10  54.81 

SUN. 

26 

o  20  38.75 

9.086 

2    H      5'5 

58.79 

5  47-39 

0.770 

o  14  51.36 

Mon. 

27 

o  24  16.81 

9.086 

2  37  35-i 

58.67 

5  28.90 

0.770 

o  18  47.91 

Tues. 

28 

o  27  54.88 

9.087 

3     i     1.6 

+58.53 

5  10.41 

0.769 

o  22  44.47 

Wed. 

29 

o  31  32.99 

9.089 

3  24  24.7 

58.38 

4  5i-97 

0.767 

o  26  41.02 

Thur. 

3° 

o  35  n.  16 

9.092 

3  47  44-o 

58.22 

4  33-59 

0.764 

o  30  37-57 

Frid. 

o  38  49.41 

9.096 

4  10  59.0 

58.04 

4  J5-29 

O.760 

o  34  34-12 

Sat. 

32 

o  42  27.77 

9.101 

N.  4  34     9.7 

+57-84 

3  57.09 

0.756 

o  38  30.68 

NOTE.—  The  semidiam&er  for  mean  noon  may  be  assumed  the  same  as  that  for  apparent  noon. 

Diff.  for  i  Hoar, 

The  sign  +  prefixed  to  the  hourly  change  of  declination  indicates  that  south  declinatioas  are 

+  98.856s. 

decreasing,  north  declinations  increasing. 

(Table  m.) 

176 


II. 


JANUARY,  1900. 


AT  GREENWICH  MEAN  NOON. 

THE  SUN'S 

i 

4 

§ 

Equation  of 

Sidereal 

a 

Time, 

Time, 

9 

o 

to  be 

or 

*5 

.c 

Subtracted 

Right  Ascension 

•3 

•3 

Apparent 

Diff.  for 

Apparent 

Diff  for 

from 

Diff.  for 

of 

8 

fr 

a 

Right  Ascension 

i  Hour. 

Declination. 

i  Hour. 

Mean  Time. 

i  Hour. 

Mean  Sua 

Mon. 

i 

h      m       s 

18  46  23.63 

s 
11.045 

S.23       I    23.1 

+12.24 

m       s 

3  40-17 

1.190 

h     m       s 

18  42  43.46 

Tues. 

2 

18  50  48.55 

11.031 

22    56    15.4 

13-39 

4    8-53 

1.175 

1  8  46  40.02 

Wed. 

3 

18  55  13.12 

11.015 

22    50   40.4 

14-53 

4  36.54 

1.159 

18  50  36.58 

Thur. 

4 

18  59  37.28 

10.998 

22  44  38.1 

+15.66 

5     4-15 

1.141 

18  54  33-13 

Frid. 

5 

19     4     1.04 

10.980 

22    38      8.7 

16.78 

5  31-35 

1.  122 

18  58  29.69 

Sat. 

6 

19     8  24.33 

10.961 

22    31    12.6 

17.89 

5  ,58-08 

I.I03 

19     2  26.25 

SUN. 

7 

19  12  47.13 

10.940 

22    23    49.8 

+19.00 

6  24.33 

1.082 

19     6  22.81 

Mon. 

8 

19  17     9.41 

10.918 

22    l6      0.6 

20.10 

6  50.05 

1.  060 

19  10  19.36 

Tues. 

9 

19  21  31.17 

10.895 

22     7  45-3 

21.  l8 

7  15-25 

1.037 

19  14  15.92 

Wed. 

10 

19  25  52.36 

10.870 

21  59     4-0 

+22.25 

7  39-87 

I.OI4 

19  18  12.48 

Thur. 

1  1 

19  30  12.94 

10.845 

21  49  57.1 

23-31 

8     3-90 

0.990 

19    22      9.04 

Frid. 

12 

19  34  32.93 

10.819 

21    40    24.8 

24-37 

8  27.34 

0.964 

19  26   5.59 

Sat. 

13 

19  38  52.30 

10.793 

21    30    27.3 

+25.41 

8  50.14 

0-937 

19  30   2.15 

SUN. 

14 

19  43  ii.  01 

10.766 

21    2O      4.9 

26.43 

9  12.30 

0.910 

19  33  58-71 

Mon. 

15 

19  47  29.07 

10.738 

21     9  18.0 

27-45 

9  33-8i 

0.882 

19  37  55-26 

Tues. 

16 

19  51  46.45 

10.710 

20    58      6-7 

+28.46 

9  54-63 

0.854 

19  41  51.82 

Wed. 

17 

19  56     3.15 

10.681 

20    46    31.4 

29.46 

10  14.77 

0.825 

19  45  48.38 

Thur. 

18 

20     .0   19.15 

10.651 

20  34  32-5 

30.44 

10  34.21 

0-795 

19  49  44.94 

Frid. 

19 

20     4  34.42 

10.621 

20    22    IO.I 

+31.41 

10  52.93 

0.765 

19  53  41.49 

Sat. 

20 

20     8  48.98 

10.591 

20     9  24.7 

32.37 

ii   10.94 

0-735 

19  57  38.05 

SUN. 

21 

20    13       2.79 

10.560 

19  56  16.6 

33-31 

ii  28.18 

0.704 

20      I    34.61 

Mon. 

22 

2O    17    15.86 

16.529 

19  42  46.1 

+34-23 

ii  44.70 

0.672 

20    5  31.16 

Tues. 

23 

2O   21    28.17 

10.497 

19  28  53.5 

35-14 

12      0.45 

0.641 

20    9  27.72 

Wed. 

24 

20    25    39.7? 

10.465 

19  14  39-3 

36.04 

12    15.44 

0.609 

20    13    24.28 

Thur. 

25 

20    29    50.49 

10.433 

19     o     3.8 

+36.92 

12    29.66 

0.577 

20    17    20.83 

Frid. 

26 

20  34     0.48 

10.400 

1  8  45    7.4 

37-78 

12    43.09 

0.544 

2O   21    17.39 

Sat. 

27 

20  38     9.69 

10.367 

18  29  50.4 

38.62 

12    55.74 

0.511 

20    25    13.94 

SUN. 

28 

20  42  18.10 

10.334 

18  14  13.3 

+39-45 

13      7-59 

0.478 

20   29    10.50 

Mon. 

29 

20   46    25.71 

10.300 

17  58  16.5 

40.27 

13    18.64 

0.444 

20  33     7.06 

Tues. 

30 

20  50  32.49 

10.266 

17  42     0.4 

41.07 

13    28.88 

0.410 

20  37    3.61 

Wed. 

31 

20  54  38.46 

10.232 

17  25  25.4 

41.85 

13    38.29 

0.376 

20   41      0.17 

Thur. 

32 

20    58    43.61 

10.198 

S.  17     8  31.9 

+42.61 

13    46.89 

0.341 

20  44  56.72 

NOTE.—  The  semidiameter  for  mean  noon  may  be  assumed  the  same  as  that  for  apparent  noon. 

Diff.  for  i  Hour, 

The  sign  +  prefixed  to  the  hourly  change  of  declination  indicates  that  south  declinations  are 

decreasing 

(Table  III.) 

177 


JANUARY,  1900, 


III. 


AT  GREENWICH  MEAN  NOON. 

Mean  Time 
of 
Sidereal  Noon 

Day  of  the  Month. 

I 

3 

1 

sx 

THE  SUN'S 

Logarithm 
of  the 
Radius  Vector 
of  the 
Earth. 

Diff.  for 
i  Hour. 

TRUE  LONGITUDE. 

DiflE.  for 
i  Hour. 

LATITUDE. 

A 

A' 

I 
2 

3 

2 

3 

152.96 
152.96 
152.96 

4-0.26 
0.40 
0.50 

9.9926699 
9.9926694 
9.9926706 

-  0.6 

+    O.I 

0.8 

h     m       s 

5  16  24.56 
5  12  28.65 
5     8  32.74 

280  40      8.7 
28l    41    20.1 
282   42    31.4 

39  51-2 
41     2.3 
42   13.4 

4 

I 

4 
5 
6 

283  43  42.4 
284  44  53.1 
285  46     3.5 

43  24.3 
44  34.8 
45  45-o 

I52-95 
152.94 
152.92 

4-0-59 
0.65 
0.67 

9.9926734 
9.9926780 
9.9926845 

+  1.5 

2-3 

3-i 

5     4  36-83 
5     o  40.91 
4  56  45-0° 

7 

8 

9 

7 
8 

9 

286  47  13.3 
287  48  22.7 
288  49  31.5 

46  54-7 
48     3-9 
49  12.5 

152.90 
152.88 
152.85 

+  0.65 
O.6o 
0.51 

9.9926929 
9.9927034 
9.9927163 

+  3-9 
4.9 
5-9 

4  52  49-og 
4  48  53-i8 
4  44  57-27 

10 

ii 

12 

10 

ii 

12 

289  50  39.7 
290  51  47.3 
291  52  54.4 

50  20.6 

51  28.0 

52  34-9 

152.83 
152.81 
152.79 

+  0.42 
0.30 
0.17 

9.9927317 
9.9927495 
9.9927699 

+  6.9 
8.0 
9.0 

4  41      1-36 
4  37     5-44 
4  33     9-53 

*3 

H 
15 

13 
H 
15 

292  54     0.8 
293  55     6.7 

294    56    1  2.  1 

53  41-2 
54  46-9 
55  52.r 

152.76 
152.74 
152.71 

+  0.04 
—  O.og 
O.2O 

9.9927930 
9.9928190 
9.9928476 

+  10.1 

W*3 

12.5 

4  29  13.62 
4  25   17.71 
4  21   21.80 

16 

17 
18 

16 

17 
18 

295  57  16.9 

296    58    21.2 

297  59  25.0 

56  56-7 
58     0.9 

59     4-5 

152.69 
152.67 
152.65 

-0.31 

0-39 
0.45 

9.9928790 
9.9929131 
9.9929500 

+13.6 
14.8 
15-9 

4  17  25.89 
4  J3  29.98 
4     9  34-07 

19 
20 

21 

19 

20 
21 

299    o  28.3 
300    1-31.2 

301       2    3.3.5 

o     7.7 

i   10.4 

2    12.6 

152.63 
152.61 
152.59 

—  0.48 
0.48 
0.46 

9.9929894 
9.9930314 
9.9930759 

+17.0 
18.0 
19.0 

4     5  38.16 
4     i   42.24 
3  57  46.33 

22 

23 
24 

22 
23 
24 

302     3  35-4 
303     4  36-7 
304     5  37-5 

3  14-3 
4  15-5 
5  16.1 

152.57 
152.54 
152.52 

-0,41 

0-34 
0.23 

9.9931227 

9-993I7I9 
9.9932232 

+20.  o 
20.9 

21.8 

3  53  50-42 
3  49  54-51 
3  45  58.60 

3 

27 

25 
26 
27 

305     6  37.8 
306     7  37-5 
307     8  36.5 

6  16.2 
7  15-7 
8  14.6 

152.50 
152.48 
152.45 

—  O.I  I 
-f-  0.02 
0.15 

9.9932765 
9.9933316 
9-9933886 

4-22.6 

23.4 
24.1 

3  42     2.69 
3  38     6.78 
3  34  10.87 

28 
29 
3° 
31 

28 

29 
30 

31 

308     9  34.8 
309  10  32.4 
310  ii  29.0 

311    12    24.7 

9  12.8 

10    10.2 

II     6.7 

12      2.2 

152.42 
152.38 

152.34 
152.30 

+  0.29 
0.44 

o-55 
0.63 

9.9934471 

9-993507I 
9.9935684 
9.9936309 

+24.7 
25.3 
25.8 
26.3 

3  30  14-96 
3  26  19.05 
3  22  23.14 
3  18  27.23 

32 

32 

312    13    19.2 

12    56.6 

152.25 

+  0.68 

9.9936948 

+26.9 

3  H  3I-32 

NOTE.—  The  numbers  in  column  A  correspond  to  the  true  equinox  of  the  date;  in  column  A'  to  the 

Diff.  for  i  Hour, 
-9a.82g6. 
(Table  II.) 

178 


IV. 


JANUARY,  1900. 


GREENWICH  MEAN  TIME. 

THE  MOON'S 

J3 

c 

0 

5 
1 

SEMIDIAMETER. 

HORIZONTAL  PARALLAX. 

UPPER  TRANSIT. 

AGE. 

0 

>> 
B 

Noon. 

Midnight. 

Noon. 

Diff.  for 
i  Hour 

Midnight. 

Diff.  for 
i  Hour. 

Meridian  of 
Greenwich. 

Diff.  for 
i  Hour. 

Noon. 

i 

,       ., 

. 

„ 

, 

„ 

b      ra 

m 

d 

I 

16  19.6 

16  23.4 

59  48-5 

+1-33 

60     2.8 

+  1.05 

6 

29-5 

2 

16  26.4 

16  28.3 

60  13.6 

0.75 

60    2O.7 

+0.44 

o  57-9 

2.43 

O.g 

3 

16  29.2 

16  29.2 

60  24.1 

+0.13 

60    23.8 

-0.17 

i  55-o 

2-33 

1.9 

4 

16  28.1 

16  26.2 

60    20.0 

-0.45 

60    13.0 

-0.70 

2   49.6 

2.22 

2.9 

5 

16  23.6 

16  20.2 

60       3.2 

0.92 

59  50-9 

I.IO 

3  4*-9 

2.14 

3-9 

6 

16  16.3 

16  12.0 

59  36.7 

1.25 

59  20.9 

1.36 

4  32.8 

2.IO 

4-9 

7 

16     7.4 

16     2.6 

59     4-o 

-1.44 

58  46-4 

-1.48 

5  23.1 

2.10 

5-9 

8 

*5  57-7 

15  52-8 

58  28.4 

1.50 

58    10.2 

1.50 

6  13.9 

2.13 

6.9 

9 

15  47-9 

15  43-i 

57  52-2 

1.49 

57  34-5 

1.46 

7     5-6 

2.18 

7-9 

10 

15  38.4 

15  33-8 

57  17-2 

-1.43 

57     0.3 

-1.38 

7  58-5 

2.22 

8.9 

ii 

15  29-3 

15  25.0 

56  44.0 

J-34 

56  28.2 

1.29 

8  52.2 

2.24 

9.9 

12 

15  20.9 

15  16.9 

56  13-0 

1.24 

55  58.4 

1.19 

9  45-8 

2.22 

10.9 

13 

15   i3-i 

15     9-5 

55  44-4 

-1.14 

55  3i-i 

-1.09 

10  38.4 

2.16 

11.9 

H 

15     6.0 

J5     2.7 

55  18.3 

1.03 

55     6.2 

0.98 

ii  29.1 

2.06 

12.9 

15 

H  59-6 

H  56.7 

54  54-8 

0.92 

54  44-2 

0.85 

12    17.2 

1.95 

13-9 

16 

H  54-1 

H  5i-7 

54  34-5 

-0.77 

54  25-7 

-0.68 

13      2.7 

1.85 

14.9 

*7 

14  49.6 

14  47.9 

54  '8.1 

°-58 

54  u-8 

0.47 

13    46.1 

1.76 

15-9 

18 

14  46.5 

H  45-6 

54     6.9 

0.34 

54     3-5 

-O.2I 

14   27.8 

I.7I 

16.9 

19 

14  45.2 

14  45.2 

54     i-9 

—0.06 

54     2.1 

+0.10 

15      8.5 

1.69 

17.9 

20 

H  45-9 

14  47.1 

54     4-4 

+0.28 

54     8.8 

0.46 

15  49-3 

I.7I 

18.9 

21 

14  48.9 

H  5i-4 

54  15-5 

0.65 

54  24.6 

0.85 

16  30.9 

1.76 

19.9 

22 

H  54-5 

H  58.3 

54  36-1 

+1.06 

54  50-1 

+  1.27 

17  14-3 

1.86 

20.9 

23 

15     2.8 

15     7-9 

55     6.5 

1.48 

55  25.4 

1.67 

1  8     0.4 

1.99 

21.9 

24 

15  13-7 

15  20.0 

55  46-5 

1.85 

56     9-9 

2.  02 

18  49.8 

2.14 

22.9 

25 

15  26.9 

15  34-2 

56  35-i 

+2.17 

57     i-9 

+2.29 

19  43.0 

2.29 

23-9 

26 

15  4i-8 

15  49-7 

57  30-0 

2.37 

57  58.8 

2.41 

20    39-5 

2.42 

24.9 

27 

15  57-6 

16     5.4 

58  27.8 

2.40 

58  56-4 

2.34 

21    38.5 

2.49 

25.9 

28 

16  12.9 

16  19.9 

59  23-9 

+2.22 

59  49-6 

+2.04 

22    38.3 

2.48 

26.9 

29 

16  26.2 

16  31.6 

60  12.8 

I.8l 

60  32.8 

1-52 

23  37-3 

2-43 

27.9 

30 

16  36.0 

16  39.3 

60  49.0 

I.lS 

61     0.9 

+0.80 

6 

28.9 

31 

16  41.3 

16  41.9 

61     8.2 

+0.40 

61   10.6 

o.oo 

o  34.5 

2.34 

0.4 

32 

16  41-3 

16  39-4 

61     8.3 

-0.39 

61     1.3 

-0.75 

I  29.8 

2.26 

1.4 

179 


JANUARY,  1900. 


VII 


GREENWICH  MEAN  TIME. 

THE  MOON'S  RIGHT  ASCENSION  AND  DECLINATION. 

Hour. 

Right 

Ascension. 

Diff  for 
i  Minute. 

Declination. 

Diff.  for 
i  Minute. 

Hour. 

Right 
Ascension. 

Diff.  for 
i  Minute. 

Declination. 

Diff.  for 
I  Minute. 

TUESDAY  9. 

THURSDAY  n. 

0 

h     m       » 
2     5     1.09 

2.2544     N.l6    38    10.6 

9.442 

o 

h    m       8 
3  55     9-21 

» 
2.3237 

N.22     4  39.6 

3*944 

i 

2     7   16.41 

8.2563 

16  47  34.2 

9-343 

i 

3  57  28.65 

2.3242 

22     8  32.5 

3.818 

2 

2     9  3I-85 

2.258? 

16  56  51.8 

9.243 

2 

3  59  48-12 

2.3247 

22    12    17.8 

3*692 

3 

2  ii  47-39 

2.2600 

17     6     3.4 

9.142 

3 

4     2     7-62 

2.3252 

22  15  55-5 

3.566 

4 

2  14     3.05 

2.2619 

17  15     8.9 

9.040 

4 

4    4  27.14 

2.3254 

22    19    25.8 

3.440 

5 

2    16    18.82 

2.2637 

17  24     8.2 

8.937 

5 

4.    6  46.67 

2.3257 

22   22   48.4 

3.3I3 

6 

2  18  34.70 

2.2656 

17  33     1-4 

8.835 

6 

9     6.22 

2.3260 

22    26      3.3 

3.186 

7 

2   2O   50.69 

2.2675 

17  41  48.4 

8.731 

7 

ii  25.79 

2.3262 

22   29    IO.7 

3.059 

8 

2   23      6.8O 

2.2693 

17  50  29.1 

8.627 

8 

13  45-36 

2.3262 

22   32    10.4 

2.932 

9 

2   25   23.01 

2.2712 

17  59     3-6 

8.521 

9 

16    4.93 

2.3262 

22  35    2.5 

2.805 

10 

2   27   39.34 

2.2730 

18     7  31.6 

8.414 

10 

18  24.50 

2.3262 

22  37  47.0 

8.677 

ii 

2   29    55.77 

2.2748 

'8  15  53-3 

8.307 

ii 

4  20  44.07 

2.3261 

22   40   23.8 

2-549 

12 

2   32    12.32 

2.2767 

18  24     8.5 

8.199 

12 

4  23     3-63 

2.3259 

22   42    52.9 

2.422 

*3 

2  34  28.98 

2.2785 

18  32  17.2 

8.091 

13 

4  25  23.18 

2.3257 

22  45  14.4 

2.295 

«4 

2  36  45-74 

2.2802 

18  40  19.4 

7.982 

M 

4  27  42.71 

2.3254 

22  47  28.3 

2.167 

«5 

2  39     2.61 

2.2820 

18  48  15.1 

7.873 

15 

4  30     2.23 

2.3251 

22  49  34-5 

2.039 

16 

2   41    19.58 

2.2837 

18  56     4.2 

7.76a 

16 

4  32  21.72 

2.3246 

22  51  33-o 

I.9« 

»7 

2  43  36.66 

2.2856 

19     3  46-6 

7.651 

17 

4  34  41.18 

2.3241 

22  53  23.9 

1.784 

18 

2  45  53.85 

2.2873 

19  ii  22.4 

7-539 

18 

4  37    0.61 

2.3235 

22  55     7-1 

1.657 

«9 

2   48    11.14 

2.2890 

19  18  51-4 

7-427 

19 

4  39  20.00 

2.3229 

22    56   42.7 

1.529 

20 

2   5°   28.53 

2.2907 

19  26  13.7 

7.315 

20 

4  41  39.36 

2.3222 

22    58    10.6 

1.401 

21 

2   52   46.02 

2.2923 

19  33  29.2 

7.201 

21 

4  43  58.67 

2.32:4 

22  59  30.8 

1.273 

22 

2  55     3-6: 

2.2940 

19  40  37.8 

7.086 

22 

4  46  17-93 

2.3207 

23     0  43-4 

1.  147 

23 

2  57  21.30 

2.2956 

N.I9  47  39-5 

6.971 

23 

4  48  37-15 

2.3197 

N.23     i  48.4 

1.019 

WEDNESDAY  10. 

FRIDAY  12. 

0 

2  59  39.08 

2.2972 

N.I9  54  34-3 

6.856 

0 

4  50  56-30 

2.3187 

N.23     2  45-7 

0.892 

i 

3     i  56-96 

2.2987 

20       I    22.2 

6.741 

I 

4  53  I5-40 

2.3177 

23     3  35-4 

0.764 

2 

3     4  14-93 

2.3003 

20     8     3.2 

6.624 

2 

4  55  34-43 

2.3166 

23     4  17-4 

0.637 

3 

3     6  33.00 

2,3018 

20    I4    37.1 

6.507 

3 

4  57  53-39 

s-3'54 

23     4  51-9 

0.511 

4 

3     8  5i-i5 

2.3032 

2O   21       4.0 

6.389 

4 

5    o  12.28 

«-3Ma 

23     5  18.7 

0.384 

5 

3  'I     9-39 

2.3047 

2O   27    23.8 

6.271 

5 

5     2  31.09 

2.3128 

23     5  38.o 

0.258 

6 

3  13  27.72 

2.3061 

20  33  36-5 

6.152 

6 

5     4  49-82 

2.3115 

23     5  49-7 

0.132 

7 

3  15  46-12 

2.3074 

20  39  42.1 

6.033 

7 

5     7     8.47 

2.3102 

23     5  53-8 

+  0.006 

8 

3  18     4.61 

2.3087 

20  45  40.5 

5.913 

8 

5     9  27.04 

2.3087 

23     5  50-4 

-  0.120 

9 

3  20  23.17 

2.3100 

20    51    31.7 

5-793 

9 

5  "  45-51 

2.3070 

23     5  39-4 

0.246 

10 

3    22   41.81 

2.3113 

20  57  15.7 

5.672 

10 

5  M     3-88 

«.3053 

23     5  20.9 

0.371 

ii 

3  25    0.53 

2.3125 

21       2    52.4 

5-55* 

ii 

5  16  22.15 

2.3037 

23     4  54-9 

0.495 

12 

3  27  19.31 

2.3136 

21     8  21.9 

5.43I 

12 

5  18  40.32 

2.3019 

23     4  21.5 

0.619 

13 

3  29  38.16 

2.3147 

21    13    44.1 

5.308 

13 

5  20  58.38 

2.3001 

23     3  40.6 

0-744 

14 

3  3i  57.o8 

2.3158 

21   18  58.9 

5.186 

14 

5  23  16.33 

2.2982 

23     2  52.2 

0.868 

15 

3  34  16.06 

2.3168 

21    24      6.4 

5.064 

15 

5  25  34.16 

2.2962 

23     *  56-4 

0.992 

16 

3  36  35-io 

2.3178 

21  29     6.6 

.941 

16 

5  27  51.87 

2.2942 

23     o  53.2 

1.115 

17 

3  38  54-20 

2.3187 

21  33  59-3 

.8.7 

17 

5  30    9-46 

2.2921 

22  59  42.6 

1.237 

18 

3  4i   13-35 

2.3196 

21  38  44.6 

.693 

18 

5  32  26.92 

2.2899 

22  58  24.7 

1.360 

19 

3  43  32.55 

2.3204 

21  43  22.5 

.569 

19 

5  34  44.25 

8.2877 

22  56  59-4 

1.482 

20 

3  45  51.80 

2.3212 

21  .47  52.9 

•445 

20 

5  37     i-44 

2.2853 

22  55  26.8 

1.604 

21 

3  48  ii.  10 

2.3220 

21   52  15.9 

.320 

21 

5  39  18.49 

2.2830 

22  53  46-9 

1.725 

22 

3  50  30-44 

2.3226 

21   56  31.3 

.194 

22 

5  4i  35.40 

2.2807 

22   51    59.8 

1.845 

23 

3  52  49.8i 

2.3231 

22      0    39.2 

4-069 

23 

5  43  52-17 

2.2782 

22  50     5-5 

1.965 

24 

3  55    9-21 

2.3237 

N.22     4  39«6 

3-944 

24 

5  46     8.79 

2.2757 

N.22   48     4.0 

2.085 

180 


VI. 


FEBRUARY,  1900. 


GREENWICH  MEAN  TIME. 

THE  MOON'S  RIGHT  ASCENSION  AND  DECLINATION. 

Hour. 

Right 
Ascension. 

Diff.for 

t  Minute. 

Declination. 

Diflf.  for 
i  Minute. 

Hour. 

Right 

Diff.for 
i  Minute. 

Declination. 

Diff.for 
i  Minute. 

MONDAY  5. 

WEDNESDAY  7. 

i    m       s 

s 

O           t           n 

• 

h     m        s 

8 

O             *              M 

* 

0 

51   20.62 

a.  295* 

N.is  26  59.8 

10.261 

o 

3  42  30*05 

8.3283 

N.2I  30  28.1 

4-685 

i 

53  38-36 

2.2962 

15  37  12.4 

10.  159 

i 

3  44  49-75 

2.3283 

21  35     5-4 

4.558 

2 

55  56-16 

8.2971 

15  47  18.9 

10.057 

2 

3  47     9-45 

2.3283 

21  39  35-1 

4*432 

3 

58  14.01 

2.2981 

15  57  19-3 

9*954 

3 

3  49  29.15 

2.3282 

21  43  57-2 

4.305 

4 

o  31-93 

a.  2991 

16     7  13.4 

9.850 

4 

3  5i  48.84 

2.3280 

21  48  11.7 

4*179 

5 

2  49.90 

2.3001 

16  17     1.3 

9*745 

5 

3  54     8.51 

3.3378 

21  52  18.7 

4.052 

6 

5     7-94 

3.3012 

16  26  42.8 

9.639 

6 

3  56  28.17 

2.3376 

21  56  18.0 

3.935 

7 

7  26.04 

2.3021 

16  36  18.0 

9-533 

7 

3  58  47-82 

3.3272 

22      O      9.7 

3.798 

8 

9  44.19 

2.3030 

16  45  46.8 

9.427 

8 

4     *     7-44 

2.3268 

22      3    53.8 

3.672 

9 

12      2.40 

2.3040 

16  55     9.2 

9.319 

9 

4     3  27.04 

2.3264 

22      7    30.3 

3.544 

10 

14    20.67 

2.3050 

17     4  25.1 

9.210 

10 

4     5  46-61 

3.3259 

22    10    59.2 

3.417 

ii 

16  39.00 

2.3059 

17  »3  34-4 

9.100 

ii 

4     8     6.15 

2.3254 

22    14    20.4 

3.290 

12 

18  57.38 

2.3069 

17  22  37.1 

8.990 

12 

4  10  25.66 

2.3248 

22    17   34.0 

3.163 

13 

21   15.83 

2.3080 

17  3i  33-2 

8.880 

13 

4  12  45.13 

2.3242 

22    20   39.9 

3-035 

14 

23  34-34 

3.3089 

17  40  22.7 

8.769 

M 

4  15     4-57 

2.3236 

22    23   38.2 

8.908 

15 

25  52-90 

2.3098 

17  49     5-5 

8.657 

»5 

4  17  23.96 

2.3228 

22    26    28.9 

•.781 

16 

28  11.52 

2.3108 

17  57  4!-5 

8.544 

16 

4  19  43-3- 

2.3221 

22    29    II.9 

S.653 

17 

30  30.20 

2.3117 

18     6  10.8 

8.431 

17 

4    22       2.6l 

2.3212 

22    31    47-3 

8.537 

18 

32  48-93 

3.3126 

18   14  33.2 

8.317 

18 

4  24  21.86 

2.3204 

22  34  15.1 

8.400 

19 

35     7.7i 

2.3135 

18  22  48.8 

8.202 

19 

4  26  41.06 

2.3195 

22   36   35.3 

a.373 

20 

37  26.55 

2.3144 

18  30  57-5 

8.087 

20 

4  29     0.20 

8.3184 

22    38    47.8 

8.145 

21 

39  45-44 

2.3153 

18  38  59-3 

7.972 

21 

4  31   19.27 

8.3173 

22  40   52.7 

3.018 

22 

42     4-39 

2.3162 

18  46  54.1 

7.856 

22 

4  33  38-28 

3.3163 

22   42    50.O 

1.892 

23 

44  23-39 

2.3170 

N.i8  54  42.1 

3-739 

23 

4  35  57-23 

2.3152 

N.22  44  39.7 

1.765 

TUESDAY  6. 

THURSDAY  8. 

o 

46  42.43 

2.3177 

tf.ig      2   22.9 

7.622 

o 

4  38  16.10 

2-3139 

N.22   46    21.8 

1.638 

i 

49     1-52 

8.3186 

19     9  56.7 

7.504 

i 

4  40  34.90 

2.3127 

22  47  56.3 

1.512 

2 

51  20.66 

3.3194 

19  17  23.4 

7-386 

2 

4  42  53-62 

2.31I3 

22  49  23.2 

1.386 

3 

53  39.85 

2.3202 

19  24  43.0 

7-267 

3 

4  45  12.26 

2.3100 

22    50   42.6 

1.260 

4 

55  59-oS 

8.3208 

19  3i  55-5 

7.148 

4 

4  47  30-82 

2.3086 

22  51  54-4 

1.134 

5 

58  18.35 

2.3215 

19  39     0.8 

7.028 

5 

4  49  49-29 

2.3071 

22    52    58.7 

1.008 

6 

3    o  37.66 

2.3222 

19  45  58.9 

6.908 

6 

4  52     7-67 

3.305J 

22  53  55-4 

o.88z 

7 

3     2  57.01 

2.3228 

19  52  49.8 

6.787 

7 

4  54  25.95 

3.3039 

22   54   44.6 

0-757 

8 

3     5  16.40 

8.3234 

*9  59  33-4 

6.666 

8 

4  56  44-14 

2.3023 

22  55  26.3 

0.632 

9 

3     7  35-82 

2.3240 

20     6     9.7 

6.545 

9 

4  59     2.23 

2.3007 

22    56      0.5 

0.508 

10 

3     9  55-28 

2.3246 

20  12  38.8 

6.424 

10 

5       I    20.22 

2.2989 

22    56    27.3 

0.384 

ii 

3  12  14.77 

8.3251 

20  19     0.6 

6.303 

ii 

5     3  38-10 

2.2971 

22    56   46.6 

0.260 

12 

3  14  34-29 

2.3256 

20  25  15.0 

6.178 

12 

5     5  55-87 

2.2932 

22    56    58.5 

0.136 

13 

3  16  53.84 

2.3260 

20    31    22.0 

6.056 

13 

5     8  I3-52 

2.2933 

22  57     2.9 

+  0.012 

14 

3  »9  i3-4i 

2.3264 

20  37  21.7 

3*933 

14 

5  10  31.06 

3.2913 

22  56  59-9 

0.1(1 

15 

3  21  33.01 

2.3268 

20  43  14.0 

5.8o8 

15 

5  12  48-48 

2.2893 

22    56    49.6 

0.233 

16 

3  23  52-63 

2.3271 

20   48    58.8 

5-685 

16 

5  15     5-78 

2.2872 

22    56    31.9 

0.357 

17 

3  26  12.26 

2.3273 

20  54  36.2 

S.56I 

J7 

5  17  22.95 

2.2851 

22  56     6.8 

0*479 

18 

3  28  31.91 

2.3277 

21      O      6.1 

3-436 

18 

5  19  39-99 

2.2829 

22  55  34-4 

0.601 

19 

3  30  51-58 

8.3279 

21     5  28.5 

5.3i* 

19 

5  21   56.90 

2.2807 

22  54  54-7 

0.733 

20 

3  33  "-26 

2.3281 

21     10    43.5 

3-187 

20 

5  24  13-67 

8.2784 

22  54     7-7 

0.843 

21 

3  35  30-95 

2.3282 

21    15    50.9 

5.061 

21 

5  26  30.31 

2.2761 

22   53    13-5 

0.963 

22 

3  37  50-65 

2.3283 

21    20    50.8 

4-936 

22 

5  28  46.80 

2.2737 

22    52    12.  1 

1.084 

23 

24 

3  40  10.35 
3  42  30.05 

2.3283 
2.3283 

21    25    43.2 
N.2I    30   28.1 

4.8u 

4.685 

23 
24 

5  3»     3-i5 
5  33  19.36 

8.3713 

8.2689 

22  5»     3-4 
N.22  49  47.6 

1.204 
1.323 

181 


XVIII. 


FEBRUARY,  1900. 


GREENWICH  MEAN  TIME. 

LUNAR  DISTANCES. 

** 

P 

Name  and  Direction 
of  Object 

Midnight 

P.L. 
of 

Diff. 

XVk 

P.L.. 

of 
Diff. 

XVIIIk 

P.L. 
of 
Diff. 

XXIk 

P.L, 
Diff. 

18 

Pollux               W. 

86  46  29 

3081 

88  15     2 

3076 

89  43  41 

3069 

91    12   28 

3063 

Regulus             W. 

49  49     o 

3051 

51   18  10 

3°44 

52  47  28 

3037 

54  16  55 

3030 

Antares             E  . 

50     7  41 

3038 

48  38  15 

3032 

47     8  42 

3026 

45  39     2 

3022 

JUPITER            E  . 

50  57  29 

3062 

49  28  33 

3056 

47  59  30 

3050 

46  30  19 

3044 

SATURN             E  . 

74  45  51 

3050 

73  16  4° 

3044 

71  47  22 

3038 

70  17  57 

3°3i 

a  Aquilae            E  . 

102  48  44 

3500 

101  28  20 

3489 

ioo     7  43 

3479 

98  46  55 

3468 

19 

Pollux               W. 

98  38  27 

3027 

100     8     6 

3020 

K»  37  54 

301  r 

I°3     7  53 

3003 

Regulus             W. 

61  46  34 

2989 

63   17     i 

2979 

64  47  40 

2969 

66  18  31 

2960 

Antares             E  • 

38     8  59 

2992 

36  38  36 

2985 

35     8     4 

2977 

33  37  23 

2971 

JUPITER            E  . 

39     2  19 

3007 

37  32  15 

•998 

36       2       O 

2989 

34  3i  34 

2980 

SATURN             E  . 

62.  48  39 

2994 

61  18  19 

2985 

59  47  48 

2976 

58  17     5 

2967 

a  Aquilae            E  • 

92     o     7 

3421 

90  38  14 

3412 

89  16  ii 

3403 

87  53  58 

3395 

20 

Regulus            W. 

73  55  57 

2906 

75  28     8 

2894 

77     o  34 

2882 

78  33  16 

2870 

JUPITER             E  . 

26  56  24 

2930 

25  24  43 

23  52  49 

2909 

22    2O   41 

2898 

SATURN             E  . 

50  40  26 

2914 

49     8  25 

2903 

47  36  10 

2891 

46     3  39 

2878 

a  Aquilae            E  . 

81     o  35 

3356 

79  37  28 

3350 

78  14  14 

3343 

76  50  52 

3337 

SUN                    E. 

109  33  56 

3275 

to8     9  15 

3262 

106  44  19 

3249 

105  19     8 

3235 

21 

Regulus             W. 

86  20  53 

2802 

87  55  18 

2788 

89  30     i 

2773 

9i     5     4 

2758 

Spica                W. 

32  18  23 

2793 

33  53     o 

2779 

35  27  56 

2763 

37     3  12 

2747 

SATURN             E  . 

38  16  56 

36  42  42 

2797 

35     8  10 

2782 

33  33  18 

2766 

a  Aquilae            E  . 

69  52  25 

3313 

68  28  29 

33" 

67     4  30 

3309 

65  40  29 

3307 

SUN                   E. 

98     9     o 

3162 

96  42     5 

3U6 

95  14  51 

3130 

93  47  18 

3114 

22 

Spica                W. 

45     4  53 

2666 

46  42  19 

2649 

48  20     8 

2631 

49  58  21 

2613 

SATURN             E  . 

25  33  48 

2686 

23  56  49 

2669 

22    19    28 

2652 

20  41  44 

2634 

«  Aquilae            E  . 

58  40  32 

3322 

57  16  46 

3330 

55  53     9 

334° 

54  29  44 

3352 

SUN                    E. 

86  24  25 

3028 

84  54  47 

3009 

83  24  45 

5989 

81  54  19 

2971 

23 

Spica                W. 

58  15  32 

2522 

59  56  H 

2504 

61  37  21 

2485 

63  18  55 

2467 

a  Aquilae            E  . 

47  37  27 

3468 

46  16  27 

3505 

44  56     8 

3548 

43  36  37 

3598 

SUN                   E. 

74  16  17 

2876 

72  43   28 

2856 

71   10  13 

2836 

69  36  32 

2817 

24 

Spica                W. 

7i  53  21 

2373 

73  37  34 

2355 

75  22  14 

4335 

77     7  22 

2317 

JUPITER           W. 

24   51    20 

2407 

26  34  45 

2387 

28  18  39 

2367 

30    3     i 

2348 

SUN                  E. 

61  41  44 

2719 

60     5  29 

2699 

58  28  48 

2680 

56  5i  4i 

2661 

25 

Spica               W. 

85  59  40 

2228 

87  47  26 

2210 

89  35  38 

2194 

91  24  15 

2177 

JUPITER           W. 

38  51  44 

2256 

40  38  49 

2238 

42    26    20 

2220 

44  *4  J7 

2203 

SUN                   E  . 

48  39  50 

2571 

47     o  15 

2554 

45  20  17 

2538 

43  39  56 

2522 

26 

Spica                W. 

TOO  33  22 

2101 

102    24    19 

2087 

104  15  38 

2074 

106     7  17 

2061 

Antares             W. 

55     5  57 

2113 

56  36  37 

2098 

58  47  39 

2085 

60  39     2 

2071 

JUPITER            W. 

53  20  ii 

2126 

55  10  31 

2112 

57     i  12 

2093 

58  52  14 

2085 

SATURN            W. 

29  40     6 

2120 

3i  30  35 

2105 

33  21  26 

2092 

35  12  37 

2079 

SUN                   E. 

35  13     9 

2457 

33  30  55 

8446 

31  48  26 

2438 

30     5  46 

2433 

27 

JUPITER            W. 

68  12     4 

2030 

70     4  51 

2021 

7i  57  52 

2013 

73  5i     6 

2005 

SATURN            W. 

44"  33  « 

2025 

46  26     7 

2017 

48  19  15 

2009 

50    12    36 

2000 

SUN                  E. 

21  31  15 

2439 

19  48  36 

2456 

18     6  21 

248l 

16  24  41 

2520 

182 


MARCH,  1900. 


AT  GREENWICH  APPARENT  NOON. 

THE  SUN'S 

j 

8 

| 

Sidereal 

Equation  of 

i 

•z. 

Semi- 

Time, 
to  be 

1 

1 

diameter 

Added  to 

•3 

"S 

Apparent 

Diff.  for 

Apparent 

Diff.  for 

Semi- 

Passing 

Apparent 

Diff.  for 

I 

.1 

Right  Ascension. 

i  Honr. 

Declination, 

diameter. 

Meridian, 

Time. 

i  Hour. 

Thur 

I 

h      m        s 

22  47  57.02 

s 
9-371 

S.  7  38  24.5 

+56.96 

16     9.26 

65-37 

m        » 

12    34.70 

.485 

Frid 

2 

22    51    41.63 

9-350 

7  15  34-2 

57,22 

16     9.02 

65-30 

12    22.8O 

.506 

Sat. 

3 

22  55  25.75 

9-330 

6  52  37-9 

57-47 

16     8.77 

65-23 

12    10.41 

.527 

SUN. 

4 

22  59     9.38 

9.310 

6  29  35.9 

+57-70 

16     8.52 

65.16 

"    57-52 

•547 

Mon. 

5 

23       2    52.53 

9.291 

6     6  28.7 

57-91 

16     8.27 

65.09 

II    44.16 

.566 

Tues 

6 

23     6  35.24 

9.272 

5  43  16.5 

58.11 

1  6     8.02 

65.03 

II    30.35 

0.584 

Wed. 

7 

23   10  17.52 

9.254 

5  *9  59-8 

+58.29 

16     7.77 

64.97 

II    l6.I2 

0.602 

Thur. 

8 

23  13  59-37 

9.237 

4  56  39-0 

58.46 

16     7.52 

64.91 

II       1.46 

0.619 

Frid. 

9 

23  17  40.83 

9.221 

4  33  H-4 

58.61 

16     7.26 

64.86 

10  46.40 

0.635 

Sat. 

10 

23    21    21.91 

9.206 

4     9  46.5 

+58.74 

1  6     7.00 

64.81 

10  30.97 

0.650 

SUN. 

ii 

23    25       2.64 

9.192 

3  46  15-5 

58.86 

16     6.74 

64.76 

10  15.19 

0.665 

Mon. 

12 

23    28    43.04 

9.178 

3  22  41.9 

58.96 

1  6     6.48 

64.71 

9  59-07 

0.679 

Tues. 

13 

23    32    23.11 

9.165 

2  59     6.1 

+59-05 

16     6.22 

64.66 

9  42-65 

0.691 

Wed. 

H 

23    36       2.90 

9.154 

2  35  28.3 

59.12 

16     5.96 

64.62 

9  25.93 

0.702 

Thur 

15 

23  39  42.42 

9.144 

2    II    48.9 

59.18 

1  6     5.69 

64.58 

9     8.95 

0.712 

Frid 

16 

23  43  21.72 

9-134 

I    48      8.4 

+59-23 

16     5.42 

64-55 

8  51-73 

0.721 

Sat. 

i7 

23  47     0.80 

9.125 

I    24    27.0 

59-26 

16     5-15 

64.52 

8  34-31 

0.730 

SUN. 

18 

23  50  39.68 

9.118 

i     o  45.0 

59.27 

*6     4.88 

64-49 

8  16.68 

0.738 

Mon. 

19 

23  54  18.38 

9.113 

o  37     2.8 

+59-27 

16    4.61 

64.47 

7  58-87 

0.744 

Tues. 

20 

23  57  56-94 

9.106 

S.   o  13  20.8 

59.26 

16     4.34 

64-45 

7  4°-93 

0.750 

Wed. 

31 

o     i   35-37 

9.101 

N.  o  10  20.7 

59.23 

1  6     4.07 

'  64.43 

7  22.85 

0-755 

Thur. 

22 

o     5  I3-7I 

9.098 

o  34     i-3 

+59-i8 

16     3.79 

64.4I 

7     4-69 

o-759 

Frid. 

23 

o     8  51.97 

9-095 

o  57  40-8 

59-12 

16     3-51 

64.40 

6  46.44 

0.762 

Sat. 

24 

0    12    30.17 

9.093 

i   21   18.6 

59-05 

16     3.24 

64'39 

6  28.14 

0.764 

SUN. 

25 

o  16     8.33 

9.092 

i  44  54.6 

+58.97 

1  6    2.96 

64.39 

6    9.80 

0.765 

Mon. 

26 

o  19  46.47 

3.092 

2      8    28.1 

58.86 

16     2.69 

64.38 

5  5i-45 

0.765 

Tues. 

27 

o  23  24.61 

9-093 

2  3i  59-i 

58.74 

16     2.41 

64.38 

5  33-09 

0.764 

Wed. 

28 

o  27     2.78 

9.094 

2  55  27.0 

+58.60 

16    2.13 

64.38 

5  14-77 

0.763 

Thur. 

29 

o  30  41.00 

9-096 

3  18  51.5 

58.44 

16     1.86 

64-39 

4  56-48 

0.761 

Frid. 

30 

o  34  19.27 

9-098 

3  42  12.2 

58.27 

16     1.58 

64.40 

4  38.23 

0-759 

Sat. 

31 

o  37  57.60 

9.101 

4     5  28.7 

58.09 

16    1.31 

64.41 

4  20.05 

0.756 

SUN. 

32 

o  41  36.01 

9.105 

N.  4  28  40.7 

+57.90 

16     1.03 

64.42 

4     1^8 

0.752 

NOTE.—  The  mean  time  of  semidiameter  passing  may  be  found  by  subtracting  o«.i8  from  the  sidereal  time. 

The  sign  +  prefixed  to  the  hourly  change  of  declination  indicates  that  south  declinations  are  decreasing;  nonb 

declinations,  increasing. 

183 


II. 


MARCH,  1900. 


AT  GREENWICH  MEAN  NOON. 

Day  of  the  Week. 

Day  of  the  Month. 

THE  SUN'S 

Equation  of 
Time, 
to  be 
Subtracted 
from 
Mean  Time. 

Diff.  for 
i  Hour. 

Sidereal 
Time, 
or 
Right  Ascension 

Mean  Sun. 

Apparent 
Right  Ascensioa 

Diff.  for 
i  Hour. 

Apparent 
Declination. 

Diff.  for 
i  Hour. 

Thur. 
Frid. 
Sat. 

I 

2 

3 

h      m       s 

22  47  55.05 

22    51    39.70 

22  55  23.86 

s 
9-371 
9-350 
9-330 

S.   7  38  36-4 
7  15  46-0 
6  52  49-6 

57-22 

57-47 

12    34.80 
12    22.90 
12    10.51 

0.485 
0.506 
0-527 

22  35  20.24 
22  39  16.80 
22  43  13-35 

SUN. 
Mon. 
Tues. 

4 
5 
6 

22  59     7-53 

23      2    50.72 

23     6  33.47 

9.310 
9.291 
9.272 

6  29  47.4 
6     6  40.0 
5  43  27.6 

+•57-70 
57-91 
58.11 

II    44.27 
1  1    30.46 

0-547 
0.566 

0.584 

22  47     9.90 

22    5I       6.45 

22  55     3.01 

Wed. 
Thur. 
Frid. 

7 

8 

9 

23  10  15.79 
23  13  57-68 
23  17  39-i8 

9-254 
9-237 
9.221 

5  20  10.7 
4  56  49-7 
4  33  24.9 

+58.29 
58.46 
58.61 

II     16.23 

ii     i-57 
10  46.51 

0.602 
0.619 
0.635 

22    58    59-56 
23       2    56.11 

23     6  52.66 

Sat. 
SUN. 

Mon. 

10 

ii 

12 

23    21    20.30 
23    25       1.07 
23    28   41.51 

9.206 
9.192 
9.178 

4     9  56.8 
3  46  25.6 
3  22  51.8 

+58.74 
58.86 
58.96 

10  31.08 
10  15.30 
9  59-i8 

0.650 
0.665 
0.679 

23  10  49.22 

23  14  45-77 
23  18  42.32 

Tues. 
Wed. 
Thur. 

13 

H 

23    32   21.63 
23    36      1-47 

23  39  41.04 

9.165 
9.154 
9.144 

2  59  15-7 
2  35  37-7 

2    II    58.0 

+59-05 
59.12 
59.18 

9  42.76 
9  26.04 
9     9.06 

0.691 
0.702 
0.712 

23    22    38.88 

23  26  35-43 
23  30  31-98 

Frid. 
Sat. 
SUN. 

16 

17 

18 

23  43  20.38 
23  46  59.50 
23  50  38.42 

9.134 
9.125 
9.118 

I    48    17.2 
1    24    35-5 

i     o  53.2 

+59-23 
59.26 
59-27 

8  51-84 
8  34-4i 
8  16.78 

0.721 

0.730 
0.738 

23  34  28.54 
23  38  25.09 
23  42  21.64 

Mon. 
Tues. 
Wed. 

19 

20 
21 

23  54  17.17 

23  57  55-77 
o     i  34.25 

9.112 
9.106 
9.101 

o  37  10.7 

S.    o  13  28.4 
N.  o  10  13.4 

+59-27 
59.26 
59-23 

7  58.98 
7  22.95 

0-744 
0.750 
0-755 

23  46  18.19 
23  50  14.74 
23  54  "-So 

Thur. 
Frid. 
Sat. 

22 

23 
24 

o     5  12.63 
o     8  50.93 

O    12    29.18 

9.098 
9-095 
9-093 

0  33  54-3 
o  57  34-i 

I    21     I2'.2 

+59-I8 
59.12 
59.05 

7    4-78 
6  46-53 
6  28.22 

0-759 
0.762 
0.764 

23  58     7-85 

O      2      4.40 

o     6     0.96 

SUN. 

Mon. 
Tues, 

25 
26 

27 

o  16    7.39 
o  19  45.58 
o  23  23.77 

9.092 
9.092 
9-093 

i  44  48.5 
2     8  22.3 
2  31  53-6 

+58.97 
58.86 
58.74 

6     9.88 
5  5i-52 
5  33-i6 

0.765 
0.765 
0.764 

o     9  57-51 
o  13  54.06 
o  17  50.61 

Wed. 
Thur. 
Frid. 
Sat. 

28 
29 
3° 
31 

o  27     1.99 
o  30  40.25 
o  34  18.56 
o  37  56.94 

9.094 
9.096 
9.098 
9.101 

2  55  21.8 
3  1  8  46-7 
3  42     7-7 
4     5  24-5 

+58.60 
58.44 
58.27 
58.09 

5  14-83 
4  56.54 
4  38-29 
4  20.  1  2 

0.763 
0.761 
0.759 
0.756 

O   21    47.16 

o  25  43.72 
o  29  40.27 
o  33  36.82 

SUN. 

32 

o  41  35.40 

9.105 

N.  4  28  36.8 

+57-90 

4    2.03 

0.752 

o  37  33-37 

NOTE.—  The  semidiameter  for  mean  noon  may  be  assumed  the  same  as  that  for  apparent  noon. 
The  sign  +  prefixed  to  the  hourly  change  of  declination  indicates  that  south  declinations  are 
decreasing  ;  north  declinations,  increasing. 

Diff.  for  i  Hour, 
(Table  III.) 

184 


II. 


APRIL,  1900. 


AT  GREENWICH  MEAN  NOON. 

Day  of  the  Week. 

| 

i 

THE  SUN'S 

Equation  of 
Time, 
to  be 
Subtracted 

Diff.  for 

Sidereal 
Time, 

Right  Ascension 
Mean  Sun. 

Apparent 
Right  Ascension. 

Diff.  for 
i  Hour. 

Apparent 
Declination. 

Diff.  for 
j  Hour. 

Added  to 
Mean  Time. 

Mon.' 
Tues. 

2 

3 

h      m       s 

o  41  35.40 
o  45  13.96 
o  48  52.63 

9.105 
9.109 
9.114 

N.  4  28  36.8 
4  51  44.2 
5  14  46-3 

+57.90 
57.69 
57-47 

4     2.03 

3  44-04 
3  26.15 

0.752 
0.748 
0-743 

h      m        s 

o  37  33-37 
o  41   29.93 
o  45  26.48 

Wed. 
Thur. 
Frid. 

4 
5 
6 

o  52  31.44 
o  56  10.39 
o  59  49.49 

9.120 
9.126 
9-133 

S  37  42-7 
6     o  33.2 
6  23  17.3 

+57-23 
56.97 
56.70 

3     8.40 

2    50.8l 
2    33.36 

0-737 
0.731 
0.724 

o  49  23.03 

o  53  19-59 
o  57  16.14 

Sat. 
SUN. 
Mon. 

7 

8 

9 

i     3  28.79 
i     7     8.27 
i   10  47.97 

9.141 
9.150 
9.160 

6  45  54-7 
7     8  25.1 
7  30  48-2 

+56.41 
56.11 

55.80 

2    16.09 

i  59-02 

I  42.17 

0.716 
0.707 
0.697 

i     i   12.69 
i     5     9-24 
i     9     5.8o 

Tues. 
Wed. 
Thur. 

10 

ii 

12 

i   14  27.91 
i   18     8.ii 

I    21    48.58 

9.170 
9.181 
9-193 

7  53     3-7 
8  15  ii.  i 
8  37  10.3 

+55-48 
55-15 
54.80 

I  25.57 

I     9.21 

0    53-13 

0.687 
0.676 
0.664 

1   *3     2.35 
i   16  58.90 

I    20   55.46 

Frid. 
Sat. 
SUN. 

Mon. 
Tues. 
Wed. 

13 

H 

16 

18 

i  25  29.35 
i  29  10.43 
i  32  51-84 

i  36  33.60 
i  40  15.72 
i  43  58.24 

9.206 
9.219 
9-233 

9.248 
9.264 
9.280 

8  59     0.8 
9  20  42.4 
9  42  14-7 

10     3  37.4 
10  24  50.1 
10  45  52.7 

+54-43 
54.05 

+53-24 
52.82 
52-39 

o  37-34 
o  21.86 
o     6.72 

0.651 
0.638 
0.624 

0.609 
0-593 
0.576 

I  24  52.01 
I  28  48.56 

i  32  45-12 

i   36  41-67 
i  40  38.22 
i  44  34.78 

o     8.07 
o  22.50 
o  36.54 

Thur. 
Frid. 
Sat. 

19 

20 
21 

i  47  41.17 
i   51   24.51 
i  55     8.29 

9.297 
9-3I5 
9-334 

ii     6  44.7 
ii  27  25.9 
ii  47  55.8 

+51.94 
51.48 
51.01 

o  50.16 

I   3-38 

I   16.15 

0.559 
0.541 
0.523 

i  48  31.33 
i  52  27.88 
i  56  24.44 

SUN. 

Mon. 
Tues. 

22 
23 

24 

i  58  52.53 
2     2  37.24 
2     6  22.42 

9-353 
9-373 
9-393 

12     8  14.2 

12    28    20.7 
12    48    15.0 

+50.52 
50.02 
49.50 

I  28.46 

I  40.32 
I  51.68 

0.504 
0.484 
0.464 

2      0   20.99 

2     4  J7-55 
2     8  14.10 

Wed. 
Thur. 
Frid. 

y 

27 

2    10      8.09 
13    54.26 
17   40.94 

9-413 
9-434 
9-455 

13     7  56.8 

13    27    25.7 
13    46   41.4 

+48.97 
48.43 
47.87 

2      2.56 
2    12.95 
2    22.82 

0.443 
0.422 
0.401 

2    12    10.65 

2  16     7.21 

2    20      3.76 

Sat. 
SUN. 
Mon. 

28 
29 

y- 

21    28.13 
25    15.84 
29      4-07 

9.476 
9.498 
9.520 

14     5  43-5 
14  24  31.7 
14  43     5.6 

+47.29 
46.70 
46.10 

2    32.18 
2    41.03 
2    49.36 

0.380 
0.358 
0.336 

2    24      0.32 
2    27    56.87 
2    31    53-42 

Tues. 

3* 

2    32    52.82 

9-542 

N.is     i  25.0 

+45-49 

2    57.16 

0.314 

2  35  49-98 

NOTE.—  The  semidiameter  for  mean  noon  may  be  assumed  the  same  as  that  for  apparent  noon. 
The  sign  +  prefixed  to  the  hourly  Change  of  declination  indicates  that  north  declinations  are 
increasing. 

Diff.  for  i  Hoar, 
+  9'.8565. 
(Table  III.) 

185 


II. 


AUGUST,  1900. 


AT  GREENWICH  MEAN  NOON. 

THE  SUN'S 

Equation  of 

j 

-a 

Time, 

Sidereal 

1 

S 

to  be 
Subtracted 

Time, 

a 

1 

from 

Right  Ascension 

•g 

Apparent 

Diff.  for 

Apparent 

Diff.  for 

Added  to 

Diff.  for 

of 

>, 

Right  Ascension. 

i  Hour. 

Declination. 

i  Hour. 

Mean  Time. 

i  Hour. 

Mean  Sun. 

a 

Q 

Wed. 

I 

h      m        s, 

8  44  41.30 

s 
9.716 

N.i8°    5     i.i 

-37.61 

6     8.12 

s 
0.140 

h      m       s 

8  38  33.18 

Thur. 

2 

8  48  34.18 

9.690 

17  49  49-7 

38.34 

6     4.44 

o.i  66 

8  42  29.74 

Frid. 

3 

8  52  26.44 

9.664 

17  34  21.0 

39-05 

6    0.15 

0.191 

8  46  26.29 

Sat. 

4 

8  56  1  8.08 

9-638 

17  18  35.2 

-39-75 

5  55-23 

0.217 

8  50  22.85 

SUN. 

5 

9     o     9.10 

9.613 

17       2    32.6 

•  4°-45 

5  49-70 

0.243 

8  54  19-4° 

Mon. 

6 

9     3  59-5i 

9.587 

16  46  13.5 

4I.I3 

5  43-55 

0.269 

8  58  15.96 

Tues. 

7 

9     7  49-31 

9.562 

16  29  38.3 

-41.80 

5  36-80 

0.294 

9     2  12.51 

Wed. 

8 

9  "   38-51 

9-537 

1  6  12  47.1 

42.46 

5  29.44 

0.319 

9     6     9.07 

Thur. 

9 

9  15  27.11 

9-5I3 

15  55  4°-3 

43.10 

5  21.48 

0-344 

9  10     5.62 

Frid. 

10 

9  19  15-12 

9.489 

15  38  18.2 

-43-73 

5  12.94 

0.368 

9  14     2.18 

Sat. 

1  1 

9  23     2.56 

9.465 

15    20    41.0 

44-35 

5     3-82 

0.392 

9  i7  58.73 

SUN. 

12 

9  26  49.43 

9.442 

15       2    49.1 

44.96 

4  54-14 

0.415 

9  21  55-29 

Mon. 

13 

9  30  35-75 

9.419 

14  44  42.7 

-45.56 

4  43-91 

0.438 

9  25  51-84 

Tues. 

H 

9  34  21.54 

9-397 

14    26    22.  0 

46.15 

4  33-14 

0.460 

9  29  48.40 

Wed. 

15 

9  38     6.80 

9-375 

H     7  47-5 

46.72 

4  21.85 

0.482 

9  33  44-95 

Thur. 

16 

9  4i  5*-55 

9-354 

13  48  59-4 

-47-28 

4  10.04 

0.503 

9  37  4*-50 

Frid. 

17 

9  45  35-79 

9-333 

13  29  57.9 

47.83 

3  57-73 

0.524 

9  41  38.06 

Sat. 

18 

9  49  *9-54 

9.3I3 

13  10  43.5 

48.36 

3  44-93 

0-544 

9  45  34-6i 

SUN. 

19 

9  53     2.81 

9-293 

12    51     16.4 

-48.88 

3  31-64 

0.564 

9  49  31.17 

Mon. 

20 

9  56  45.61 

9.273 

12    31     37.0 

49-38 

3  17-89 

0.583 

9  53  27.72. 

Tues. 

21 

10     o  27.94 

9.254 

12    II    45.6 

49.88 

3     3-67 

0.602 

9  57  24.28 

Wed. 

22 

10     4     9.83 

9.235 

II    51    42.6 

-50.36 

2  49.00 

0.620 

10     i  20.83 

Thur. 

23 

10     7  51.27 

9.217 

II    31    28.2 

50.83 

2    33-89 

0.638 

10     5  17.38 

Frid. 

24 

10  ii   32.28 

9.199 

11     II       2.8 

51-28 

2    18.34 

0.656 

10    9  13.94 

Sat. 

25 

10  15   12.86 

9.182 

10  50  26.8 

-51.72 

2      2.37 

0.674 

10  13  10.49 

SUN 

26 

10  18  53.03 

9.165 

10  29  40.5 

52.14 

i   45-99 

0.691 

10  17     7.04 

Mon. 

27 

10    22    32.8l 

9.149 

10     8  44.3 

52-55 

i   29.21 

0.707 

10   21       3.60 

Tues. 

28 

10  26  12.19 

9-133 

9  47  38-4 

-52.94 

i   12.04 

0.723 

10  25     0.15 

Wed. 

29 

10  29  51.20 

9.118 

9  26  23.2 

53-32 

o  54.50 

0.738 

10  28  56.70 

Thur. 

30 

10  33  29.86 

9.104 

9     4  59-i 

53-69 

o  36.60 

0-753 

10  32  53.26 

Frid. 

31 

10  37     8.16 

9.090 

8  43  26.3 

54.04 

o  18.35 

0.767 

10  36  49.81 

Sat. 

32 

10  40  46.13 

9.076 

N.  8  21  45.2 

-54-38 

o     0.23 

0.781 

10  40  46.36 

NOTE.—  The  semidiameter  for  mean  noon  may  be  assumed  the  same  as  that  for  apparent  noon. 

Diff.  for  i  Hour, 

The  sign  —  preBxed  to  the  hourly  change  of  declination  indicates  that  north  declinations  are 

+  9'.8565. 

decreasing. 

(Table  IIL) 

186 


SEPTEMBER,  1900 


III. 


AT  GREENWICH  MEAN  NOON. 

THE  SUN'S 

• 

| 

0 

1 

i 

TRUE  LONGITUDE. 

Logarithm 
of  the 
Radius  Vector 

Mean  Time 

•5 

i 

Diff.  for 

LATITUDE. 

of  the 

Diff.  for 

of 

I 

s 

A 

y 

t  Hour. 

Earth. 

I  Hour. 

Sidereal  Noon 

i 

244 

158  34  13-5 

33  23.3 

145-25 

—  0.18 

0.0038298 

-44.4 

h     m       s 
13    17      2.70 

2 

245 

159    32    20.2 

31  29.9 

145.31 

—  0.06 

0.0037217 

44-9 

13    13      6.80 

3 

246 

160  30  28.3 

29  37.9 

M5-37 

+  0.07 

0.0036125 

43-4 

13     9  10.89 

4 

247 

161  28  37.8 

27  47-3 

M5-43 

-f-  O.2O 

0.0035024 

-45-9 

13     5  I4-98 

5 

162  26  48.7 

25  58-2 

145.49 

0.28 

0.0033914 

46.3 

13     i  19.07 

6 

249 

163    25      1.2 

24  10.6 

M5-55 

o-35 

0.0032799 

46.6 

12    57    23.17 

7 

250 

l64   23    15.3 

22    24.5 

145-62 

+  0.40 

0.0031678 

-46.8 

12  53  27.26 

8 
9 

251 
252 

165    21    3I.O 

1  66  19  48.4 

2O   40.1 

18  57-5 

145.69 
M5-77 

0.42 
0.40 

0.0030554 
0.0029427 

46-9 
47-o 

12  49  31.35 
12  45  35-45 

10 

253 

167  18     7.8 

17  16.7 

145-85 

+  0.35 

0.0028297 

-47-1 

12  41  39.54 

ii 

254 

1  68  1  6  29.0 

15  37-9 

M5-93 

0.27 

0.0027166 

47-2 

12  37  43.64 

12 

255 

169  14  52.3 

14     i.o 

146.01 

0.15 

0.0026032 

47-3 

12  33  47-73 

13 

256 

170  13  17.6 

12    26.3 

146.10 

+  0.03 

0.0024894 

-47-5 

12    29    51.82 

H 

257 

171   ii  45.2 

10  53-7 

146.19 

—  O.IO 

0.0023751 

47-7 

12    25    55.91 

15 

258 

172  10  14.9 

9  23.4 

146.28 

0.23 

O.OO226O3 

48.0 

12    22      O.OI 

16 

259 

173     8  46.8 

7  55-2 

146-37 

-0.35 

0.0021448 

-48-3 

12   18     4.10 

17 

260 

174     7  20.9 

6  29.2 

146.46 

0.47 

0.0020285 

48.6 

12    14      8.19 

18 

261 

175     5  57-2 

5     5-5 

146.56 

0.56 

O.OOI9II4 

49.0 

12    10    12.29 

19 

262 

176    4  35.8  v 

3  43-9 

146.65 

—  0.63 

0.0017934 

-49-4 

12    6  16.38 

20 

263 

177    3  16.5 

2    24.6 

146.74 

0.68 

0.0016743 

49.8 

12      2    20.48 

21 

264 

178     i  59-3 

i     7-3 

146.83 

0.70 

0.0015543 

50.2 

II    58    24.57 

22 

265 

178  60  44.2 

59  52-0 

146.91 

—  0.70 

0.0014332 

—50.6 

ii  54  28.66 

23 

266 

179  59  31.0 

58  38.8 

147.00 

0.66 

O.OOI3II2 

51.0 

ii  50  32.76 

24 

267 

1  80   58    20.0 

57  27.7 

147.08 

0.61 

O.OOII882 

5'-4 

ii  46  36.85 

25 

268 

181  57  10.9 

56  r8.5 

147.16 

-0.53 

0.0010642 

-51.8 

ii  42  40.94 

26 

269 

182  56     3.7 

55  "-3 

147.24 

0.44 

0.0009394 

52.1 

ii  38  45.04 

27 

270 

183  54  58-4 

54     5-9 

M7-32 

0.33 

0.0008137 

52.4 

ii  34  49-13 

28 

271 

184  53  55-o 

53     2.4 

147-39 

—  0.21 

0.0006872 

-52.7 

ii  30  53.22 

29 

272 

185  52  53-4 

52    0.7 

147-47 

—  0.08 

0.0005603 

33-o 

ii  26  57.32 

30 

273 

186  51  53.6 

51     0.8 

147-54 

+  0.04 

0.0004328 

53-2 

ii  23     1.41 

31 

274 

187  50  55-5 

50    2.6 

147.62 

4-0.17 

0.0003049 

-53-3 

ii  19    5-5° 

NOTE.—  The  numbers  in  column  A  correspond  to  the  true  equinox  of  the  date  ;  in  column  A'  to  the 

Diff.  for  t  Hour. 

—  OT829-6. 

mean  equinox  of  January  od.o. 

(Table  II.) 

187 


II. 


NOVEMBER,  1900. 


AT  GREENWICH  MEAN  NOON. 

THE  SUN'S 

Day  of  the  Week. 

£ 

8 

•?. 

1 
I 

Equation  of 
Time, 
to  be 
Added  to 
Mean  Time. 

Diff.  for 

Sidereal 
Time, 

Right  Ascension 
Mean  Sun. 

Apparent 
Right  Ascensioa 

Diff.  for 
i  Hour. 

Apparent 
Declination. 

Diff.  for 
i  Hour. 

Thur 
Frid. 
Sat 

i 

2 

3 

h      m        s 

14  24  57-33 
14  28  52.67 
14  32  48.80 

9.790 
9.823 
9-856 

S.  I4    22    58.5 
I4    42       8.4 
15        I        4.0 

-48.20 
47.61 
47.01 

16  18.76 
16  19.97 
1  6  20.40 

0.067 
0.034 
0.00  1 

14   41    16.09 

14  45  12.64 
14  49     9.20 

SUN. 
Mon. 
Tues. 

4 
5 
6 

14  36  45.72 
14  40  43.46 

14  44  42.01 

9.889 
9.923 
9.958 

15    19    44.8 
15    38     10-5 

15  56  20.6 

-46-39 
45-75 
45.09 

16  20.03 
16  18.85 
16  16.85 

0.032 
0.065 

0.100 

H  53     5-75 
14  57     2.31 
15     o  58.86 

Wed. 

Thur. 
Frid. 

7 

8 

9 

14  48  41.40 
14  52  41.62 
14  56  42.70 

9-993 
10.028 
10.063 

16  14  14.9 
16  31  52.8 
16  49  14.1 

-44.42 
43-73 
43-03 

16  14.02 
16  10.35 
16     5-83 

0-135 

0.170 

0.206 

15     4  55-42 
15     8  51.97 

15    12    48.53 

Sat. 
SUN. 
Mon. 

10 

1  1 

12 

15     o  44.63 

15     4  47-43 
15     8  51.09 

10.099 
10.135 
10.171 

17     6  18.2 
17  23     4.8 
17  39  33-6 

-42.31 
41-57 
40.82 

16     0.45 
15  54,2  1 
15  47.10 

0.242 
0.278 
0.314 

15  16  45.08 

15    20   41.64 

15  24  38.19 

Tues. 
Wed. 
Thur. 

13 

14 

J5 

15  12  55.63 
15  17     1.03 
15  21     7.30 

10.207 
10.243 
10.279 

17  55  44-i 
18  ii   35.9 
18  27     8.6 

-40.05 
39-26 
38.46 

15  39-12 
15  30.27 
15  20.56 

0.351 
0.387 
0.423 

15  28  34.75 
15  32  31-30 

15  36  27.86 

Frid. 
Sat. 
SUN. 

16 

17 

18 

15  25  14.43 
15  29  22.42 
15  33  31.26 

10.315 
10.351 
10.386 

18  42  21.8 
18  57  15-1 
19  ii  48.1 

-37-64 
36.80 
35-95 

15     9.98 

H  58.55 
14  46.27 

0.459 
0.494 
0.529 

15  40  24.42 

15  44  20.97 
15  48  17.53 

Mon. 
Tues. 
Wed. 

19 

20 
21 

15  37  40.94 
15  41-  51.46 
15  46     2.79 

10.421 
10.455 
10.489 

19  26     0.5 
19  39  5^-8 
19  53  21.8 

-35.08 
34.19 
33-29 

H  33-14 
14  19.18 
14     4.40 

0.564 

0.598 

0.632 

15  52  14.08 
15  56  10.64 
16     o     7.20 

Thur. 
Frid. 
Sat. 

22 
23 

24 

15  50  14.94 
15  54  27.87 
15  58  41.59 

10.522 
10.555 
10.587 

20     6  29.8 
20  19  15.8 

20    3I    39.2 

-32.37 
31-44 
30.50 

13  48.82 
13  32-43 
13  15-27 

0.666 
0.699 
0-731 

16     4     3-75 
16     8     0.31 
16  ii  56.86 

SUN 
Mon. 
Tues. 

25 
26 
27 

16     2  56.07 
16     7  11.29 
16  ii  27.24 

10.619 
10.649 
10.678 

20  43  39.8 
20  55  17.2 
21     6  31.0 

-29-54 
28.57 
27-58 

12  57-35 

12    38.69 
12     19.30 

0.762 
0.792 
0.822 

16  15  53-42 
16  19  49.98 
16  23  46.53 

Wed. 
Thur. 
Frid. 

28 
29 
30 

16  15  43.88 

l6    20       1.22 

16  24  19.22 

10.707 
10.736 
10.764 

21     17    2O-9 
21    27    46.7 

21   37  48.0 

-26.58 
25.56 
24.54 

II    59-21 
II    38.43 
II     16.99 

0.851 
0.880 
0,907 

16  27  43.09 
1  6  31   39.65 
16  35  36.20 

Sat 

31 

16  28  37.86 

10.790 

S.2i  47  24.5 

-23.51 

10  54.90 

0.933 

16  39  32.76 

NOTE.—  The  semidiaraeter  for  mean  noon  may  be  assumed  the  same  as  that  for  apparent  noon. 
The  sign  —  prefixed  to  the  hourly  changs  of  declination  indicates  that  soutn  declinations  are 

Diff.  for  i  Hour, 
4-9'.8s65. 

mcreasmg. 

(Table  III.) 

188 


VENUS,   1900. 


GREENWICH  MEAN  TIME. 

JANUARY. 

FEBRUARY. 

| 

Apparent 
Right 
Ascension.  . 

Var.  of 
R.A. 
for  i 
Hour. 

Apparent 
Declination. 

Var.  of 
Decl. 
for  i 
Hour. 

Meridian 

A 

7. 

Apparent 
Right 

Var.  of 
R.A. 

Hour. 

Apparent 
Declination. 

Var.  of 
Decl. 
for  i 

Meridian 

•5 

Passage. 

0 

Passage. 

1 

Noon. 

Noon. 

MM*. 

Noon. 

1 

Noon. 

Noon. 

Noon. 

Noon. 

t 

20  39  23.93 

+12.819 

-20    9  51.7 

+46.64 

156-8 

I 

23     746.06 

+H.S33 

-6  58  20.4 

+75-82 

2  22.9 

2 

20  44  30.90 

12.763 

19  5°  55-3 

48.06 

i  57-9 

2 

23  12  .5.26 

11.199 

6  27  55.5 

76.24 

2  23.4 

3 

204936.51 

12.706 

1931  25.0 

49-45 

i  59.1 

3 

23  16  43-65 

11.167 

5  57  20.9 

76.63 

2  23.9 

4 

20  54  40.75 

12.648 

19  ii  21.7 

50.82 

2     O.2 

4 

23  21  II.  26 

11.136 

5  26  37.4 

76.99 

224.5 

5 

20  59  43-61 

12.5 

y> 

18  50  46.0 

52.15 

2      1.3 

5 

23  25  38.14 

11.106 

4  55  45-8 

77-31 

2  25.0 

6 

21    4  45.08 

+12.532 

-18  29  38.8 

+53-45 

2     2.4 

6 

23  30     4-31 

+11.077 

-4  24  46.7 

+77-60 

225.5 

7 

21     94,5-15 

12.474 

18    8    0.8 

54-72 

2     3-5 

7 

23  34  29.81 

11.050 

3534J.I 

77-86 

2  25.9 

8 

21  14  43.82 

12.4 

5 

174552.7 

55-95 

2     4-5 

8 

23  38  54-68 

11.024 

3  22  29.6 

78.09 

2  26.4 

9 

21  19  41.08 

12-357 

17  23  15-4 

57-15 

2    5-5 

9 

23  43  18.95 

11.000 

2  51   13.0 

78.19 

2  26.9 

10 

21  24  36.95 

12.299 

17    o    9.6 

58-32 

2     6.5 

10 

2347  42.66 

10.978 

2  19  52.O 

78.46 

227.3 

ii 

21  2931.42 

+12.241 

-16  36  36.1 

+59-46 

2    7-5 

ii 

2352    5-85 

+10.957 

-I  48  27.4 

+78.60 

227-7 

12 

21  34  24.51 

I2.I 

3 

16  12  35.7 

60.57 

2     8.4 

12 

23  56  28.56 

10.937 

I   l6  59.8 

78.70 

2  28.2 

13 

21  39  16.22 

12.1 

f 

1548  9.2 

61.64 

2    9-3 

13 

o    o  50.82 

10.919 

o  45  29.9 

78.77 

2  28.6 

14 

2144  6.57 

12.0 

0 

15  23  17.3 

62.68 

2  1O.2 

M 

o    5  12.68 

10.903 

-o  13  58.6 

78.82 

2  29.1 

15 

21  48  55.57 

12.014 

14  58  0.9 

63.69 

2  II.  I 

'3 

o    934.17 

10.889 

+o  17  33.5 

78.84 

2  29.5 

16 

21  5343-25 

+  11.959 

-14  32  20.7 

+64-66 

2  II.9 

16 

0  1355-34 

+10.876 

+049    5.7 

+78.83 

2  29.9 

17 

21  58  29.61 

11.905 

14    6  17.5 

65.60 

2  12.8 

17 

o  18  16.22 

10.865 

I  20  37.2 

78.79 

230.3 

18 

22      3  14.70 

11.852 

13  39  52.0 

66.51 

2  13.6 

18 

0  22  36.86 

10.856 

1  52  7.4 

78.72 

230.7 

19 

22     7  58.53 

ii.  8 

0 

13  13    5-i 

67-39 

2  14.3 

J9 

o  26  57.29 

10.848 

2  23  35.6 

78.62 

2  3I.I 

20 

22  12  41.11 

11.749 

12  45  57.6 

68.23 

2  15.1 

20 

o  31  17.54 

10.841 

2  55    i.o 

78.49 

2  31-5 

21 

22  17  22.48 

+11.699 

-12  18  30.2 

+69.04 

2.5.8 

21 

o  35  37-66 

+10.836 

+3  26  23.0 

+78.33 

2  3'-9 

22 

22  22     2.67 

11.650 

ii  5043.7 

69.83 

2  16.6 

22 

o  39  57-69 

10.833 

3  57  4°-8 

78.14 

2  32-3 

23 

22  26  41.70 

11.602 

II  22  38.8 

70.58 

2  17-3 

23 

o  44  17.66 

10.832 

4  28  53.8 

77-92 

2  32.6 

24 

22  31  19.60 

11-556 

10  54  16.6 

71.29 

2  l8.0 

24 

o  48  37.62 

10.832 

5    o    1.2 

77-68 

233-0 

25 

22  35  56-41 

11.511 

10  25  37.6 

71-97 

2   18.7 

25 

o  52  57-59 

10.833 

5  31    2.4 

77.41 

233-4 

26 

224032.15 

+11.4 

3 

-  9  56  42-5 

+72.62 

2  19-3 

26 

o  57  17.62 

+10.836 

+6    i  56.6 

+77." 

233-8 

27 

22  45    6.86 

11.426 

927  32.1 

73-24 

2  19-9 

27 

i    i  37-74 

10.840 

6  32  43.2 

76.77 

2  34-2 

28 

22  49  40.57 

11.385 

858    7-4 

73-82 

2  20.5 

28 

i    557-96 

10.846 

7    3  21.4 

76.40 

2  34-6 

29 

22  54  13-32 

11.345 

8  28  29.0 

74-37 

2  21.  1 

29 

i  10  18.33 

10.853 

7  33  50-5 

76.01 

2  35-o 

30 

225845.13 

11.306 

7  58  38-0 

74.89 

2  21-7 

30 

i  14  38.88 

10.860 

8    4    9.8 

75-59 

2  35-4 

31 

23    3  16.03 

+  11.269 

-  7  28  34.9 

+75-37 

222-3 

31 

i  18  59.62 

+10.869 

+8  34  18.5 

+75-14 

235-8 

32 

23    746.06 

+11.233 

-  6  58  20.4 

+75-82 

2  22.9 

32 

i  23  20.59 

+10.879 

+9    4  16.0 

+74-65 

2  36.2 

Day  of  the  Month. 

1st. 

6th.    llth.   16th 

21st.   26th.   31st. 

Day  of  the  Month.          6th. 

10th.       15th.       20th. 

25th. 

Semidiameter 
Hor.  Parallax 

LS 

593    6.03    6.14 
6.15    6.25    6.35 

' 

6.24    6.37    6.50 
6.47    6.59    6.73 

Semidiameter  .     .       6.6 
Hor.  Parallax               6.8 

3      6.78      6.95      7.15 
1      703      7-19      7-37 

7-31 

7-57 

i 

NOTE.- 

-The  sign  +  indicates  north  declinations  ;   the  sign  —  indicates  south  declinations. 

189 


14  DAY  USE 

RETURN  TO  DESK  FROM  WHICH  BORROWED 

ASTRON-MATH-STAT. 


This  book  is  due  on  the  last  date  stamped  below,  or 

on  the  date  to  which  renewed. 
Renewed  books  are  subject  to  immediate  recall. 


MAY  2  3 1994 


LD  21-1001 


LD  21-40m-10,'65 
(F7763slO)476 


General  Library 

University  of  California 

Berkeley 


M298787 


V/Vyy 


